It was not "the nations" but rather the entrenched elites of those nations who feared the lasting reforms of the French Revolution. They feared these reforms because the reforms would upset the power structure in their countries.
The French Revolution resulted in a system that was more open to people of all classes. All people (at least all men) were supposed to be equal before the law. Rich people could join the elite of society even if they were not born to aristocratic families. Money came to be worth more than a title.
This was upsetting to the elites of many European countries. These were countries whose systems had been based on aristocratic privilege for centuries. They did not want their systems to be overthrown and for them to have to give up power to those whom they saw as their inferiors.
Most people, particularly those in power, do not welcome change. This is why the elites of these nations feared the ideas of the French Revolution.