I think its important to remember that when the French revolution happened France wasn't immediately invaded by foreign powers.

The core danger of the French revolution to the rest of Europe was the wave of political unrest that was sweeping across Europe. Very briefly the increased awareness of the disparity of wealth and the clamour for political reform meant riots and unrest throughout Europe. And of course a successful revolution in France could have dire consequences elsewhere, added to which France was by far the biggest single power in Europe at the time.

The wars of the revolution became the Napoleonic wars in time, but they're more accurately the various Coalition wars.

The first coalition war didn't start until 1793, and the aim was to contain an ever more turbulent and militaristic France. The short version is that it was beaten, and France gained some territories in the Rhineland and established the Batavian republic in the Netherlands.

Of course this was after hostilities broke out, and integral to the coalitions existence was the close relationship of European monarchs at the time. The HRE at the time (Leopold II) was Marie Antoinette's brother for instance. This helped to bind together the monarchies to restore the French royalty and preserve their own way of life.

Obviously the atrocities done during the revolution and the aggressive authoritative stance of the revolutionary government added fuel to the fire so to speak.

If you want to know more the Oxford history of the French revolution by William Doyle is a good place to start, and Revolutionary Europe 1789-1815 by Henry Stephens gives a good overview of Europe as a whole.