A lot of languages in Europe and America have Latin and Greek roots, so what caused different languages to be formed from those two when they worked fine for communication?
Also, how do different languages gain traction and people who are willing to speak a whole entire new language rather than stick with what they know?
I can understand language changing over time due to culture and the like, for example Australia, UK, and USA all speak the same language but have small differences in vocab that makes communication a bit more difficult.
I'm sure you'll get a highly detailed answer soon from a true expert, but the long story short is that languages vary regionally based on distance and isolation. All languages change over time, and with sufficient distance / barriers, they will start to diverge from each other even when they have a common origin. Being in constant communication helps blend differences, so that the languages stay "similar" over time rather than drifting apart. Thus, villages and cities tend to have their own tiny bits of flavour, groups of villages and cities will have similarities, and each part of a large country might exist on a continuum where one village is similar to the next, but if you skipped 10 villages down the road you start to notice major differences, broadly similar to the biological conception of the "Ring species".
The second thing to consider is that the having a single unified language for huge groups of people is somewhat modern, and directly related to the existence of centralized states and education systems. Languages were far less standardized in the past, and it was not until a few short hundred years ago that some of them began to be taught in a single, universal, "correct" form. The modern French language might be called "Parisian", the modern Italian language is what they were speaking in the Northern parts of Italy, rather than what they were speaking in the south. There are places in Europe to this day where the people are taught the national language in school, but speak a local dialect at home. This regionalization of languages is the "normal" state of being, so to speak, and public education systems must work to overcome that.
This is less pronounced in some parts of the world such as Australia or the United States where near infinite regional dialects did not get transmitted to begin with, and there were not many hundreds of years of divergence prior to the formation of strong national governments and easy lines of communication. However, even there we see large variation in accent and pronunciation that could theoretically diverge into entirely new languages if an impenetrable barrier suddenly sprung up and separated two halves of the country for a millennium.