You’re absolutely right that France was not in a great place at the onset of the Revolutionary Wars. France did make strides in revitalizing their economy (or at the very least they stopped the bleeding) although it was the stunning success of their military that paved the way for economic recovery and made it possible to fend off such powerful opponents in the first place.

Just to sum up just how bad France was doing economically at the onset of the first coalition war in 1792, let’s do a quick recap. France was already facing massive debt before the revolution even began. The main reason Louis XVI even called the Estates General was to get approval to levy more taxes to help pay this debt off. In 1791, France lost control of the most lucrative colony in the world when Saint Domingue, modern day Haiti, was cut off by a massive slave revolt. In 1789, France tried issuing a new paper currency backed by gold seized Catholic Church property. At the onset of the war, the value of this currency plummeted as it was pumped out to try and pay down the national debt. I could go on, but the situation doesn’t get much brighter until the end of the 1790s.

What France lacked in economic wherewithal, they made up for with military prowess. Specifically, the French military had a massive manpower reserve, occasional skyrocketing morale, and Napoleon Bonapart. After a rocky start, French morale soared once they secured a few key victories in 1792, the same year the first coalition war broke out. With these victories, France occupied Belgium, territories along the Rhine river, and parts of the Medditerranian coast. It’s here that we begin to see how France’s military success translated into economic stability. France declared it was going to spread its new enlightened way of doing things in all of the territories now under its control, although the territories would be obligated to pay for the French forces in the region while they sorted out all of the details. In practice, the armies were now paying for themselves, which eased the burden of the struggling French government.

The cycle of military success leading to economic stability leading to more military success swung back around to France proper, when in 1793 France opted to commit all available resources of the nation to the war effort. This was unheard of in 18th century Europe, as the last ~150 years had been defined by restrained “cabinet” wars. France took several steps to achieve this goal, culminating in the levée en Masse in 1793, which called up all unmarried men between 18–25 years of age, with others being funneled into manufacturing and agricultural work. Mobilization on this scale was simply unheard of, and gave the French an overwhelming boost to their manpower. This commitment to “total war” didn’t play out perfectly, but the end result was a massive net-positive to France’s military, and allowed them to shrug off the occasional defeat and still keep a large fighting force at the ready. David Bell has written a couple of books on Total War and the French Revolution, with The First Total War giving a particularly clear picture of the contrast between Revolutionary/Napoleonic warfare and earlier cabinet wars.

Jumping ahead a bit to 1796, we can see Napoleon really driving this cycle onwards, culminating in the French victory in the first coalition war in 1797. Napoleon was given given command of a withering French Army on the southern border near Italy and told to shore up France’s military position while the two main French armies meandered across Germany in the general direction of Austria while trying not to die of starvation, seeing as the French government had recently informed those two armies that they government no longer had the supplies or money to support them, and that they would need to live off the land (or pillage what they needed from the people who lived there). Napoleon had other plans, and quickly revitalized his army, captured half of northern Italy, and set his sights on the Austrian heartlands just on the other side of the Alps. The French government was literally figuratively bathing in all of the priceless Italian art, loot, gold, etc., that Napoleon was having his men ship back home by the cartload, and they didn’t want the gravy train to end just yet. This money was doing wonders for the still-struggling French economy, after all. They ordered Napoleon to wait in Northern Italy and strip away everything that wasn’t nailed down. Napoleon complied for a while, but it quickly became clear that he’d have a massive revolt on his hands. He figured with a few tens-of-thousands more men he could handle the revolt, but then he wouldn’t have the manpower to guard the Alps from Austrian counter-attacks, which had been rolling in at a steady pace since he’d first begun moving into Italy.

Napoleon being Napoleon, he solved both problems. Frist, he created a few “sister republics” in Italy by dissolving a handful of unlucky sovereign states and reorganizing them. He appointed puppet governments to keep milking them for all they’re worth lead them in his stead. Seizing the initiative, Napoleon rallied a chunk of his army, defeated the Last major Austrian-held fortress at Mantua, and started marching through the Alps into lightly defended Austrian territory. It was at this point that the Austrians quickly sought a ceasefire and later a peace treaty, ending the War of the First Coalition. For more detail on his Italian campaigns, Desmond Gregory’s Napoleon’s Italy and Michael Broers’ The Napoleonic Empire in Italy both give solid accounts of his exploits there in 1796–1797.

This primer in French/Napoleonic economics-by-conquest policy really just sets the stage for similar efforts at European reorganization and wealth extraction. Similar “sister” states were also created in Germany, the low countries, most of Italy, and Switzerland, while some territory was simply annexed, and other states retained their sovereignty in theory, but were bound so closely to France (or owed what little sovereignty they had to France) that they had no choice but to accept French demands for obscene amounts of economic support whenever they rolled in. Take Genoa, for example. In 1797, the Genoese sent a delegation to Paris to discuss terms of peace. While they were out, Napoleon swooped in and drew up and implemented a new constitution for the small Italian state and filled the ranks with Genoese Jacobins. Napoleon cheerfully remarked that the few non-Jacobins would be, “ripe for the shooting three weeks from now.” Napoleon rebranded the ancient Republic of Genoa to The Ligurian Republic, and as a christening gift, ordered the Genoese to loan France 800,000 Francs. See what I mean? For a cool, recently published account of what it was like to be an Italian politician in the French shadow, Erik Jacobs’ The Political Culture of the Sister Republics is worth checking out.

To sum up, France used its military and manpower advantages, aided in no small part by Napoleon, to shore up its economy, which would continue to struggle until Napoleon took major steps to stabilize it after coming to power, including doing away with the Assignat currency in favor of gold and silver. The baseline sources for the broader details of this answer are drawn from William Doyle’s Oxford History of the French Revolution and Georges Lefebvre’s handful of works on the French Revolution and Napoleon.

Edited, as always, for spelling and formatting.

Ah as with most things in history, it comes down to a pretty wide combination of factors-- both on the French side and on the part of the allies.

Economy:

Starting with the French economy, it's true that during the Revolution, France went through an evolving economic crisis. However it's not as straightforward as saying that their economy had crashed. While the assignat, the paper currency originally intended to be notes based upon the sale of nationalized Church lands, went through significant periods of devaluation and extraordinary inflation, the "war economy" of France shouldn't be discounted.

The assignats were often bypassed by trading instead trading "in-kind", however much more material to the war effort was the famous levée en masse, the mobilization of the entire country to arms. This decree, coming in 1793 in the midst of worsening French fortunes, was meant to do no less than focus the entire French nation on assisting with the war effort. It conscripted young men into the armed forces, women into nursing, children into linen making. Not to mention my favorite part of the decree, whereby old men were to take to their public squares to denounce monarchy and inspire virtue and love of country. For the economy, though, nothing acts like a shot in the arm more than a concerted war mobilization effort. In the industrial cities, factories were churning out arms, munitions, uniforms, etc., paid for by the state and staffed by urban workers. This effort propelled the nation forward during the early stages of the war, when France was still on the losing side. However by the end of 1793 and moving into 1794, France began to see themselves resurgant.

Although at the outset of war in 1792 France claimed high-minded goals about "introducing democracy" and "liberating the peoples of Europe" (two principles which Robespierre staunchly opposed, on the basis that foreign armies cannot force democracy upon the 'liberated'), the French army quickly learned that the expediency and efficiency of looting conquered provinces was a far more sure path to victory for the fatherland. France's war machine sought to "feed itself", first in the 'liberated' territories of the Dutch (aka Austrian Netherlands), then later into Italian territory during Napoleon's tenure. Seizure of food, clothing, supplies, and any other property that isn't nailed down is a very frugal way to provide for an army, in in the process of the war "feeding itself", France was given an economical break with her finances.

One last point on economy, but it's also worth noting that France ran off of credit-- a credit which was becoming increasingly difficult to secure before 1789 due to the obstinancy of the monarchy. While bankers take calculated risks in credit line extensions, there were far more risks in dealing with absolutist regimes-- especially ones that refused to call any form of elected bodies. pre-1789 France had a broken tax system: they used antiquated methods of collection, relying on unscrupulous tax farmers; they had an inadequate tax base, with a slew of exemptions for the well-off, but large burdens on those who could not pay their share; finally they had no agreed upon method to raise additional taxes, relying on years-long levees that had to constantly be reapproved. It's this last facet of the French system that was perhaps the biggest financial catalyst for the Revolution itself. There was no agreed upon system for instituting new, permanent taxation on the peoples of France. The monarch, supposedly 'absolute' in power, was in a deadlock with the parlements of France over raising permanent taxes. Though it's an entire question in and of itself, suffice it to say that the deadlock was a public relations nightmare at its core: the king and his regime was not popular enough to brute-force institute new taxes without risking widespread rebellion (led by the rancorous parlements), yet the parlements were leery of opposing too strenuously, as they did not want to reveal that they truly had no power in the matter, and were merely roleplaying as the defenders of the people. All this to say that, once the Estates General was called and it metamorphasized into the representative National Assembly, the creditors of Europe were far more willing to extend loans to the French, releiving some of the immediate burdens of running the state.

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French Military:

The French successes in the face of nearly unanimous opposition is explained in two parts, both equally important-- French committment to defense of their homeland and the principles of the Revolution, and the allied forces's (esp. Prussia and Austria) underestimation of, and lack of focus on, the French. Let's talk about the French forces first.

The French war effort was truly enormous. Few nations before had so entirely mobilized themselves for a war effort; I've seen it claimed that France was the first modern nation at least to put all of her manpower towards war. As I mentioned above, the levée en masse should not be understated. Every facet of the nation was put at the war's disposal, and while we discussed the material aspect of that in the economics section, there was also a vital patriotism that accompanied it. In Lynn Hunt's fantastic Politics, Culture, and Class in the French Revolution she goes into much more detail on the enormous effort the revolutionaries put into building a national identify. Briefly: pre-Revolutionary France was a patchwork quilt of different peoples, most of whom had been subsumed into 'France' at different times, under different treaties, many of whom spoke different dialects, or languages altogether, and whose sole unifying trait was their allegience to the same king. Well once 10 August 1792 rolls around, and that king is no longer at the center, you have to have something else to make the people of Flanders and the basque country feel as though they're citizens of the same nation. To this end the revolutionaries deployed a dizzying array of symbols, songs, iconography, etc. in an attempt to unite the 'French' together for defense of the motherland: and it worked.

One of the recurring themes of the French army was how unnerving their intense patriotism was. Several battles have the French described as wildly yelling "Vive la France!" while being bombarded with shellings. Whether apocryphal propoganda or not, the sentiment remains that while the Austrians and Prussians were repeatedly demoralized (more on that to come), the French were embolded, fighting not only for a cause they believed in (liberté, fraternité, egalité), but also defense of the homeland-- a power stimulant not to be taken lightly.

At this point perhaps you're thinking, "but weren't there men who were against the revolution within the army itself? Was the revolution really that popular?" Which is a great segue into another reason the French army could hold its own. Rewinding the clock a to the pre-Revolution, the French army was not in tip-top shape. France was under the Ségur ordinances, which stated among other things that the French command positions could only be staffed by the nobility. This was based on the old-world ideas of Frederick the Great, which thought that there was such a thing as a 'military aristocracy' whose greatness literally ran in their blood-- it could not be taught, nor bought. In reality, this meant that men of incredible talent were stuck as low-level captains (or, ironically, teachers of the men whose 'blood' made them supposedly superior) while the high commands were taken up by the nobility. Given the status of these commanders, they were almost, to a man, staunch defenders of the King, the Church, and their Noble perogatives. As the revolution picked up steam then in 1790-91, these men contituted a large part of the emigrés who fled France. By the time of war, then, the men left in the army were really only those who were-- at least on the surface-- committed to the revolution. The revolutionaries, due to ideology and necessity, filled the commanding roles in the army with men whose merit, rather than class, recommended them to the role. Many of the most prominent generals of the Napoleonic era-- to say nothing of Napoleon himself, who was 'discovered' by Robespierre's younger brother Augustin-- were just such promotions. This meant that the officer core and the rank-and-file were men committed to the revolutionary ideals, and willing to fight for not only their nation, but there very position, which would be eliminated (along with their lives, if they were prominent enough), if the monarchy was able to return to France.

While there is always more to say about the French army, let's move onto: (contd. in next comment)