During and right after the French Revolution, how did France maintain and operate such a big army all over Europe?

Complex question.

• Resources.
  France was large, rich and Populous country. Around 40 million to Austria's 20 million or Prussia's 10 Million, Russian was about the same 40 million, but well behind in wealth. And France was richer than these countries. And once France had established an empire, French advantages increased.

• Politics and Diplomacy.
  For a lot of the period France was only opposed to 1 major land power. Even in the Early period, Russia, Prussia were much more concerned with the partition of Poland than dealing with france.

• the Revolution - patriotism.
  Initially the revolutionary armies were able to be mobilised essentially by patriotism, and calling for volunteers was responded to in very large numbers, the revolution mobilised more of the French society, this did not last and conscription replaced volunteers. But in critical period the volunteers and enthusiasm

• The Empire.
  Once the Revolutionary armies occupied areas, Netherlands, Rhineland, Italy much of the costs of supporting the armies was able to be imposed on the Empire. as much as half the military expenditure of France came from the Empire.

The introduction of conscription was a direct result of the dissolution of the monarchy. All the other monarchies saw France as a threat to their way of order, as they feared similar uprisings. In March 1793, France was at war with Austria, Prussia, Spain, Britain, Piedmont and the United Provinces. A professional army was woefully inadequate for this, in terms of both manpower, and having to pay for it.

Desertion was fairly common but the sheer number of eligible males who were conscripted and slowly trained while fighting alongside veterans resulted in a huge, competent army of both NCO and COs.

However support grew due to the popular overthrow of the monarchy, the subsequent success in defending France, and the Napoleonic wars which leveraged this somewhat-bloodened army (and the spoils of war).

More directly to your question - the original text of the levee en masse gives some insight as to how this was accomplished (for example, the conversion of national buildings into barracks, etc.)

https://en.wikipedia.org/wiki/Lev%C3%A9e_en_masse#Text_of_the_French_Revolution.27s_lev.C3.A9e_en_masse

The French Convention decreed a levée en masse on 23 August 1793 to resist the anti-French coalition:

From this moment until such time as its enemies shall have been driven from the soil of the Republic, all Frenchmen are in permanent requisition for the services of the armies. The young men shall fight; the married men shall forge arms and transport provisions; the women shall make tents and clothes and shall serve in the hospitals; the children shall turn old lint into linen; the old men shall betake themselves to the public squares in order to arouse the courage of the warriors and preach hatred of kings and the unity of the Republic.

This state of total war marks a historic turning point. Never before in history had all men, women, children, and elderly in an entire country been requisitioned for war. This caught the coalition completely off guard. Armies until then - and for years to come still, in the rest of Europe - were the affair of professionals (and mercenaries). France overwhelmed the rest of Europe to the point where they had to expand their troop counts far beyond their ability to pay professional soldiers

At least a few historians (e.g. J.M. Roberts) emphasize the introduction of total war by France as one of the major breaks with the past that opened the modern era.

The French Revolution was more empowering than debilitating. Put another way, it awoke a sleeping giant.

At the time of the Revolution, France was the second most populous country in Europe (after Russia), and potentially the richest and strongest. The reason it failed to live up to this potential was the burdens placed on the French people by the various King Louis, especially the last one (XVI), and his wife, the extravagant Marie Antoinette.

The French Revolution eliminated these monarchs (and an oppressive nobility that ruled side-by-side with them) and thus freed a lot of resources for other purposes, including war. The Frenchmen were empowered by their newly-won freedoms, and their citizen-soldiers, led by a relative handful of veterans, fought bravely and well for many of the same reasons as newly empowered American revolutionaries some ten to fifteen years earlier, when led by many of these same professional soldiers, including one Marquis de Lafayette.

Put another way, the French had a natural quantitative superiority of resources against one or more coalition powers, and the French Revolution ensured that the quality of these resources would be at least equal, if not superior to the others.

The French revolution was truly revolutionary, it replaced the monarchy with a regime based on popular support. A king had to hire mercenaries, the revolutionary regime could draft patriotic citizens in a levee en masse.

(Sure, not all draftees were entirely enthusiastic. But it made a notable difference.)

In some parts of Europe, anti-monarchist patriots were inclined to support the republican/revoutionary French. In others, they were merely inclined to not give their utmost in the support of their king.