Their primary advantage during the Revolutionary period was a deep pool of manpower and popular support for the cause. As well as some good generals and the benefits of reforms that were implemented prior to the Revolution.

The Revolutionary/Napoleonic Wars are credited for bringing about the era of a, "Nation Under Arms." In other European nations and France prior to the Revolution, warfare was largely a specialized affair. The King had his army that clashed with similar armies, but for the most part the larger population was not enlisted in the war effort and objectives tended to be limited. Warfare was the province of kings.

That all changed with the Revolution in France. Along with the freedoms guaranteed to every French citizen came the duty of fighting for the nation. This concept(and a large population) allowed the French to field large armies and to raise new forces to plug gaps where they experienced setbacks via a levee en masse. At one point they had 1.5 million men in the field, which was enormous for the era.

Morale also was fairly high due to belief in the cause and the belief that they were fighting an existential battle. If they lost, the Republic would fall and a king would again rule. For other nations wars the general population at the national level wouldn't typically be as invested in the outcome. I would add that disorder and poor equipment was also a side effect of Revolutionary armies, so it wasn't all good. Still, the positive characteristics gave them an edge.

While the French Navy and cavalry were seriously weakened by the aristocratic purges of the officers, the infantry and artillery was less impacted, as they had more officers from the middle class. The artillery arm of the French was a strength even during the Revolutionary period. And of course Napoleon Bonaparte came out of this arm.

There were good French army generals during this period outside of Napoleon. Augereau, Bernadotte, and Desaix are a few who went on to very successful careers under Napoleon and who commanded during the Revolution.

French generals also hit on tactics that were fairly innovative at the time and would influence fighting throughout the Napoleonic Wars. This included heavy use of skirmisher swarms, staffed by the best soldiers, and fighting in column, which allowed large groups of relatively untrained men to be wielded effectively in combat. By column I mean a large solid block of men, more like a rectangle, that created a huge amount of mass which crashed into the opposing lines while being covered by skirmishers and artillery. This was a combination that early foes found very difficult to counter.

The corps system was another key reform for the French, but didn't really come into its own until the Napoleonic era.

There are tons of books on this, but the one I always recommend is The Campaigns of Napoleon, by Chandler. It covers in detail the Revolutionary period to contrast it with the Napoleonic era.