Latin didn't die. Spanish, French, Italian, Romanian, etc are all descended from Latin and spoken by millions of people today.
Languages change over time. With the loss of constant travel and cultural unity the spoken language in various areas will drift. So the people in what we call Spain will change how they speak as will the people in Burgundy and northern France. By drift alone you end up with new different languages.
Hi! As mentioned by others Latin has evolved into the many Romance languages like Italian, French, Spanish, Portuguese, Romanian, and so on. If you'd like more info on the transition from Latin to Italian as an example, check out a few posts in the FAQ
Latin continued to be considered the language of the areas of the former Roman Empire for a long time after the empire had fallen. It remained the language of religion and learning until the early 20th century, too. There was a standard, mostly-unchanging Latin that was taught in schools. But this wasn't really the vernacular language spoken even in Rome's heyday. It was a particularly erudite, sophisticated language, while the vernacular language spoken by most of the people was a bit different, and it changed through the centuries and across Europe over time. From it are descended all of the modern romance languages--French, Italian, Spanish, Catalan, Portuguese, and Romanian. People in the Italian peninsula considered their spoken language to be Latin, no matter how much what they spoke had changed from the Latin of 1000 years earlier. A key important change was when Dante wrote his Divine Comedy in the early 14th century, not in the standard erudite Latin but rather in the vernacular, and this contributed to the recognition that people were really speaking another language, and that language should become the standard, rather than having this veneer of Latin being the spoken language of the area.
We also should not forget the following (to quote J N Adams, whose The Regional Diversification of Latin 200 BC - AD 600 is a masterwork):
We should get away from the idea that Latin was monolithic until a very late date, when some catastrophic event caused it to 'split up', or that it only showed regional diversification from the Empire onwards.: (pg. 725).
That is, Latin had always shown dialectic and regional differences (even within Italy), sometimes hard to discern because of the paucity of evidence, but there nonetheless. North African Latin, in particular, was noted as being 'different' - certain syntactic and grammatic constructs found in Romance languages may have arisen there, though it left no modern descendants.
That is, there was a tendency for Latin to be a diverse language, as befits a living speech spread across a wide geographic area. The loss of an elite model on which to base 'proper speech' following the dissolution of the Western Empire no doubt accelerated this diversity.