I have a hard time understanding how so many languages descend from proto-indo-european.

My latin teacher explained that the proto-indo-europeans where the first people to ride horses, therefore they were able to travel very far, and they were admired by everyone they encountered because they were riding horses. he said its because of this PRESTIGE that the language expanded.

I find this to be a very dissatisfying answer.

How did proto-indo-european language spread??

my main question is this- was there a time when all peoples throughout indo-europe spoke THE SAME proto-indo-european language before the languages diverged? like how all people in the roman empire spoke the same latin before it diverged into the 5 romance languages.

or did proto-indo-european just INFLUENCE many separate languages in indo-europe?

I hope my question is clear- was proto-indo-european once ONE language that was spoken ACROSS the ENTIRETY of the lands that now have languages that descend from proto-indo-european?