The history of Latin after Rome is really interesting. My answer won't be a complete one, but given how often questions go unanswered, I'll try to give a partial answer here.

Latin stopped being used because it wasn't the language people spoke anymore.

That seems like a simplistic answer, but it's the case. Generally speaking, in the areas of Europe where Romances languages are spoken, what we see is Latin gradually evolved into different languages due to the influences of both native languages (various Gallic, Iberian, or Celtiberian languages in France and Spain, for example) and later languages that came into the regions (Frankish in France or Greek and Slavic languages around Romania). Add onto that the fact that over time language slowly changes due to regional variation and isolation and that there was no mass communication or education system to enforce linguistic uniformity. So, slowly over time, people stopped speaking Latin and started speaking Vulgar Latin, then various diverging languages that became the Romance languages. In places like England, the new languages of German invaders replaced the Latin and Celtic languages that had been there (though, like Latin, influences remained).

But, you're right, the Church used Latin. And not just the Church: most legal and scholarly writing was also done in Latin. So, Latin didn't really die, it just stopped being the day to day language of people and became a scholarly and religious lingua franca that united Catholic Europe. But, overtime, that language evolved too. Medieval Latin was different from Classical Latin, in part due to the fact that by the 4th and 5th centuries, Latin had changed from the language of Cicero and due to the fact that there were another 1000 years of evolution for the language to go through.

That brings us to the Italian Renaissance and the death of Latin. The Renaissance killed Latin in two ways: 1) it formalized it in a way that froze it; 2) it popularized vernacular writings. The second of those is the one that most people are taught: that Dante and Petrarch and Boccaccio all wrote in Italian and it showed people that they didn't have to write in Latin and bye-bye to all the stuffy, old-fashioned losers. Except that's wrong. Petrarch loved Latin and generally considered his Latin writing more important than his Italian poetry. And Latin was incredibly important to the Renaissance (most writings were still done in Latin by people who were very invested in Latin). But it is true that with the Renaissance we do see a growth of vernacular writings. Why? It's economic. By the 14th century, there has been an explosion of wealth within the mercantile classes in Italy, Flanders, and the Hanseatic cities, as well as elsewhere. Merchants have to be literate and educated, but they don't generally need to know Latin (they don't need to read Vergil or Augustine for their work). So we see the growth of a wealthy and literate class of people who generally cannot read the language that most literature is written in, and thus there's a growth of literature for that audience.

But that's only one of the ways that the Renaissance began to kill Latin, the other was formalizing it. Starting with Leonardo Bruni and continuing in debates about proper Latin through the 15th and into the 16th century, we see the development of Neo-Latin. Neo-Latin (a neologism, not what they called it at the time) was an attempt to return Latin to its classical roots away from the corrupting influence of Medieval Latin. They sought to return to classical spellings, grammar, and vocabulary, generally using Cicero's Latin as a model for what Latin ought to be. They were incredibly successful and by the 16th century, Latin had largely been brought back to the form (more or less) that it had in the first century BC. However, in doing so they had killed Latin. It ceased to be able to evolve outside of neologisms for new technologies, animals, and peoples. It was still used extensively, but it was no longer dynamic.

From then on, however, it's more of the same phenomenon that drove the explosion of vernacular literature: more people became literate but not in Latin, therefore there was both a larger market for vernacular writings and more writers who didn't know Latin, therefore Latin became more marginalized. By the nineteenth century, Latin had largely (but not entirely) died off outside of the Church. It was still taught in the secular and Protestant worlds, but it was very rarely used for writing.