I can understand nouns, particularly relating to things that are bought and sold (if someone in nice armour with a lot of money tells you they want to buy some panis, you'd probably make them their bread EXACTLY how they wanted it), but how did they enforce the usage of latin verbs? More so, I've heard it said that many poorer people in the Roman empire were speaking koine greek rather than latin in their day to day lives. And why did the languages keep elements of their own foundations, unlike in south america where so many of the native peoples now only speak spanish?
Bonus question! I've also heard that Korean, Japanese and Vietnamese have a similar relationship to ancient Chinese that French, English and Spanish have to Latin. Does anyone know if this relationship formed in the same way?
Yes, as /u/telkanuru rightly notes, it seems your question has some misunderstandings about Romance languages. These are all linguistic descendants of Latin. They didn't borrow vocabulary from Latin, they are Latin's daughters.
In the Eastern half of the Roman Empire, Koine Greek was a widespread dominant language. This seems unrelated to your main question.
Korean, Japanese and Vietnamese do not have the same relationship to ancient Chinese, except that their literature and literary language seems to have evolved in part from ancient Chinese, but they are not evolutionary descendants of Chinese itself. They belong to different language families.
Your question is a bit confusing and based on some preconceptions that don't seem right. Read this thread and let me know if you have any further questions.