I want to concentrate on the period 1792-1794, which is the initial post-revolutionary period, with context surrounding these years.

After the War of the Austrian Succession in 1750 France founded the Ecole Millitare, which in 1777 became the École des Cadets-gentilshommes. Its intention was to train not just high nobility to become officers, but also the sons of those wealthy and connected enough to be able to secure a place. While it was not meritocratic as modern West Point or Sandhurst, institutionally it was designed to foster learning, tactical knowledge, and encourage freedom of thought within the cadets. When Napoleon attended in 1784 he was assigned as an artillery cadet, which was one of the most prestigious positions due to the maths and tactical knowledge required. The French of the pre-revolutionary period understood the need to innovate militarily, and when the counter-revolutionary wars broke out there remained enough of an officer core that this innovative spirit remained. While it is also true that some of the remaining officers feared their own troops, the vast majority of the officers who remained were competent and usually well regarded by their men.

In respect of the War of the First Collation, the French had four major advantages over their enemies:

1. They were fighting for their own liberty on their own boarders with more or less secure supply lines and logistics (though Napoleon was always complaining about extortion from suppliers). This meant that while they fought on French or near-French soil they were usually adequately supplied with munitions, boots, horses, fodder, and other essential supplies. This was crucial in keeping the army together during the winter months, and allowed the French generals to ensure that when the campaigning season began they could fight any advance their opponents made.

2. France was a single nation fighting a coalition of disparate political ideologies. On the one hand Britain was a maritime merchant parliamentary democracy that feared French impingement on trade and an explosion of workers demanding rights. On the other, you had the autocratic monarchies of Prussia, Sardinia, and the Holy Roman Empire in the form of Austria. The coalition was disjointed, and while the coalition did enjoy success along the Rhine, because the lack of clear communication among the allies no one decisive blow was ever struck against the French because there was no effective co-ordination of military manoeuvres. For instance, at the end of the 1792 campaigning season Prussia withdrew back into her own boarders due to the economic pressures of the war, and none of her allies were able to press home the advantages she gained during her initial invasion.

3. Conscription. While it is true that in 1792 there was mass desertion of French troops, in 1793 the French were able to conscript 300,000 men via the levee en masse, which was more than the allies could place into the field. In 1793 the French had to fight on four fronts extending out from their boarders: The Netherlands, the Rhinelands, northern Italy, and the Spanish boarder. While there were rebellions in the Vende (southern France) which were serious in nature, due to the conscripted army the French were able to hold their own, and by the end of 1793 had pushed back the coalition.

4. Bear in mind that while Napoleon was the most high profile general to come out of France during this period, it took him until 1794 to gain command of an army. In the initial 18 months of the revolution men like Lazare Carnot (who introduced modern waging of war with mass armies and strategic planning), François Christophe de Kellermann (who commanded on the Rhine), Paul Barras (Commissioner to the French Army), and Jean-Baptiste Jourdan (who commanded the army in the north) all had a hand ensuring that the revolutionary army was well led and well organised. The aristocratic and counter-revolutionary classes may have fled, but many capable men remained in positions of authority during 1792/3 to ensure that while the reign of terror decimated the revolutionary ideals, France herself was kept safe from the first coalition.

So, in answer to your question: France managed to win the War of the First Coalition through a combination of the right men in the right places (such as Baras and Kellerman), a short logistics train, patriotic fervor to keep the coalition from succeeding, a highly trained and questioning officer corps, and dysfunction among the members of the coalition.

It should be added that in 1812/13 all of these factors would come back to haunt Napoleon on his Russian campaign, and the subsequent campaign that finally defeated him. Long supply trains, dysfunctional coalitions, and lack of patriotic cause led to his downfall.

Sources:

Napoleon the Great by Andrew Roberts

The Romanovs: 1613-1918 by Simon Sebag Montefiore

Alan Forrest, Soldiers of the French Revolution

TCW Blanning, The French Revolutionary Wars