For dftba-starkids: How did the Romance languages become so different from Latin?
To begin, let's assume that language evolution is a lot like biological evolution. This is actually quite a complicated assumption with lots of moving parts, so I'll break this down a bit first.
We really have no idea what the first language was. At some point in history, there was a community of people who all spoke mutually intelligible ideolects (the language as it exists inside an individual), and the community as a whole spoke what we now call Latin. Any one person's knowledge an understanding of their native language is different from every other person's, but yet somehow they all could (mostly) understand each other, just like today with English. The community that spoke Vulgar Latin, or vernacular Latin, was made up of normal people - cooks, business owners, children, farmers, soldiers. The community of Latin speakers spread out with the Roman Empire, and through trade routes and cultural transmission. You can find parallels to the spread of Latin in the spread of English today.
In any case, Latin speakers spread their culture, language, and government across huge swaths of Europe. As the locals adopted Latin, variation due to the original languages and simply due to individuals interpretations, among other reasons. This, in the most general terms, is how the Romance languages probably started: dialects of Vulgar Latin.
(adapted from wikimedia; green=living, red=deceased)
As you can see, there are many descendants of Latin in the Romance family.
There are many things that affect language change. Pretty much anything can play a role. The areas of linguistics that study language change include historical linguistics, typology, and various synchronic/diachronic subfields. Synchronic studies look at how certain things differ at the same moment in time, like the comparison of two living languages. Diachronic studies look at how certain things differ over time, like a feature of a language over a hundred years (or sometimes even just years).
One thing that often affects language change is semantic drift and syntactic reanalysis. For instance, Virginia Hill [x] discusses how the subjunctive may have arisen in Romanian. She posits that the particle in Romanian that used to be used as a "conditional complementizer" was reanalyzed as a subjunctive marker. (A complementizer is a word like "that" or "which" that marks a clause that functions as noun phrase, like a subject or object.) What this means, is that the particle in question occurred in places in the language where it could be mistaken for marking the subjunctive.
Why might this happen? Let's take another example, such as Portuguese. Without rewriting the whole story, Portuguese started off as a dialect of Vulgar Latin, but in combination with the evolution of the Hispanic languages and the influences of the Moors (who spoke Mozarabic). Mozarabic contributed a lot of vocabulary to Romance languages in general, but did not influence syntactic structure as much. Many of the current differences between Portuguese and it's sibling language Galician occurred as Portugal gained political distinction.
And this may be generalized to many languages throughout Europe.
To sum up the answer to your question: lots of small changes accumulated over time, divided by political powers (nations, kingdoms), and influenced by unrelated languages nearby. Many of these changes come from common people reinterpreting existing structures as other, slightly different structures.
Hill, V. The emergence of the Romanian subjunctive. [x] (pdf)
Joseph, B. (1983, 2009). The Synchrony and Diachrony of the Balkan Infinitive: A Study in Areal, General and Historical Linguistics. Cambridge University Press. [x] (Google Books, no text available)
Schulte, K. (2007). What causes adverbial infinitives to spread? Evidence from Romance. Language Sciences, 29: 512-537. [x]
* I really am not an expert in this topic, so please take this as an intro only. There may be errors in my summary that are refuted in more comprehensive literature. I hope you can take my intro as a way to start your own explorations!