TL;DR: Superior tactics, a lack of organisation amongst the First Coalition powers, and manpower.

Long(er) answer: There are three main reasons that France was able to overcome the odds and beat back the powers of Europe during the War of the First Coalition.

For one, the tactics adopted by the Revolutionary Army were unlike anything Europe had seen before, and this tied in very closely to the idea of equality amongst citizens. Most militaries outside of France during this period still made use of the rather static 'line infantry' style of combat, whereby soldiers would form rows and progressively duck down in these rows so that each could fire at the enemy. This was seen as the 'professional' way of fighting at the time, because it allowed the nobles who typically lead the armies to be firmly in charge. The Revolutionaries, being that they were doing away with the society of orders, adopted a tactic that would better suit this ethos - skirmishing. Instead of fighting in tight-knit groups of lines, soldiers would divide and make use of their mobility and environmental factors to quickly overpower their enemies and win battles.

It should be noted that France beat back the First Coalition on land, but was essentially dominated by Britain at sea. This was, for the first war, pretty immaterial, but I just thought I'd mention it.

Secondly, the powers that were fighting against France did not present much of a united front against them. Each was very much preoccupied in different areas of the country (Britain in the south, Prussia along the Rhine etc.) and this meant that France could pick each of them off individually without much expectation that another power would send help. It has been argued that the revolution that took place in Poland in 1794 was a significant factor in both preventing Prussia from deploying its full force against France and in keeping Russia out of the war entirely. It is certainly true that these two countries were more concerned about their local issues than about France.

Thirdly, you may know that mass conscription was introduced in France in 1793 with the levée en masse, which brought the French army up to a size of ~600,000-700,000 men (though this declined significantly as the war progressed). This was, as you can see, a lot of men, more than any other single country could deploy at the time, which was surely part of the reason France was so strong in this war.

There are many other reasons which could be advanced as well. Ideal historiography is not so popular anymore, but the ideas and spirit of the Revolution surely helped to boost the morale and determination of its soldiers, along with the meritocratic system introduced into the army at the time. Having the support of people in areas that they had occupied (for example the Netherlands and Switzerland) also helped and played important roles in later wars.

As for academic sources, I'm only a degree student, so my catalogue is somewhat limited compared to an expert. However, I can recommend Eric Hobsbawm's The Age of Revolution 1789-1848 as a great example of a history of this period (be aware that some of its views are quite dated and that Hobsbawm was a Marxist, though) and Jonathan Sperber's Revolutionary Europe, 1780-1850 as a more recent text. I also understand that William Simpson and Martin Jones' Europe 1783-1914 is good. There is also Robert Harvey's The War of Wars which deals specifically with the conflicts of this period, but this, whilst comprehensive, is more popular history than academic.

Edit: Spelling and grammar and I also thought I'd mention that Hobsbawm and Sperber specifically develop some of the points I made here in more detail, particularly the relation of military tactics to social systems.