Metternich was an Austrian alive during the French Revolution and Napoleonic Era. He would have been around to witness the revolution in France and its wide-reaching consequences, like involving his native Austria in a war to quell the rebellious revolutionaries and reinstate a monarch to the throne in France. Once Napoleon took control of France with a coup d'Etat in 1799, Metternich witnessed Napoleon's attempt to create a French Empire that encompassed much of mainland Europe.
When Metternich used the metaphor "when France sneezes, Europe catches a cold", he was referencing both events. When France's people rebelled against the absolute monarchy, the rest of Europe was tangled in the mess. When Napoleon seized power as dictator in France, he brought his war and political goals all around Europe- from trying to plant his brothers and sisters on the thrones of other European kingdoms to attempting to expand French boundaries as far as Moscow.
This metaphor can also be used to explain the series of liberal reforms that broke out in Europe a few decades after the French Revolution- most notably are the Revolutions of 1848, where liberal-backed revolutionaries fought back against the monarchies in their respective countries. Many countries saw attempts at political reform or revolution, including Poland, Austria, Italy, and Russia among others. These reforms were also partly inspired by the events of the French Revolution, which lends into Metternich's metaphor about how France impacts the rest of Europe.