Well, that's a tricky question.

First of all, let me remind that Latin in Imperial times was a highly diglossic language. Diglossia means that dialect spoken by educated minority was significantly different from one in use by common folk. In modern times, diglossia is prevented by universal education. Schooling system teaches since a person's early childhood only one standard of language, in which everything is spoken, read and written. A modern example of diglossia is thus difficult to find. Think of English of Shakespeare compared to English slang of some teen gangs in a poor urban area. Quite inaccurate comparison, but you get the picture.

Standardized languages did not exist in ancient times. Formal education for the majority of people was non-existent. Mass media such as printed books, movies, Internet - none of these obviously existed. Even more, direct linguistic contact between rich educated elite and the poor folk was limited.

On the other hand, Classical Latin was to a large degree fixed. Children in rich families were expected to learn Greek and Latin classical literature. It was also prestigious to speak what was viewed as "proper" language. As a result, while some changes obviously happened, their rate and direction did not match the direction taken by Vulgar Latin spoken by the majority of the population.

The only speakers of Classical Latin, in late Imperial times, were Roman elite and clergy of the Christian church. Even in these two groups, only a small fraction of people across the Empire had Latin as a first language. Majority of officials and priests came from different ethnicities and only spoke Latin as a lingua franca. When original Roman rulership in the West was replaced by other powers (Ostrogoths, Lombards etc), Classical Latin as native language fell out of favor. By the way, use of Latin in religious context and in international communication continued into and beyond the Renaissance. At that time, it was replaced by modern European languages.

Now we can answer OP's question: Classical Latin ceased to be a spoken language because it always had a comparatively small native speaker base, and when they lost their power and influence, the language died out as well. As an international language, it stopped being used because by this time a thousand or so years passed since anyone spoke it as a native language. It was far easier to adopt common languages such as English and French, which many people knew, than to force everyone to learn a long dead one.

What happened to Vulgar Latin then, when did it disappear? The answer is: it did not at all. When the Western Empire collapsed, population transfer between its former parts almost ceased, breaking the linguistic continuity. So, Vulgar Latin dialects evolved into Romance family of separate languages. By one definition, these dialects can no longer be called Latin starting from the point when their speakers could no longer understand Classical Latin, because of the reasons described above. By another definition, Vulgar Latin is still alive and well, its variants are just called differently: Italian, Spanish, French, Portuguese...

I hope my answer is helpful to OP.

Ad Infinitum: A Biography of Latin by Nicholas Ostler as well as an article by Arcadius Avellanus addresses this.

A major factor in the death of Latin was the Renaissance. When all the ancient texts were rediscovered, an obsession with all things ancient overtook most educated people. People tried to purify their Latin, to speak and write as Cicero would have spoken and written. The issue was that this was not natural. Cicero's works were composed, not spoken extemporaneously. This made it very hard to speak in Latin and write in Latin. On top of this difficulty, Latin had changed since the days of the Roman Empire. Vocabulary and grammar had changed, not enough to make the language different; well-educated people could understand Latin from the days of Caesar. Particularly in the realm of vocabulary there was much development. Almost all scientific and philosophic work had been done in Latin, and new vocabulary had been added for this purpose. The Renaissance wiped this slate clean; Erasmus wrote quite a bit on this in Ciceronianus. This does not mean that using Latin as a language ceased, but it became more difficult. The use of Latin in academia to publish new works continued; the last major work to be published exclusively in Latin was Newton's Principia Mathematica I believe.

A final nail in the coffin would be a result of this purification. When Latin became a language of reading, it became taught as a language of reading. Learning how to parse became important. Combine the natural difficulty of translating from a native tongue with the need to imitate a style is Latin a few hundred years old, and you get people reluctant to write in Latin. Furthermore, it is even harder to speak like that. As I said earlier, the style of Latin which was taught was a style which was almost always composed even when it was contemporary. So the habit of talking in Latin died, and this really killed the language; a language must be spoken to stay alive. When it was no longer spoken, its utility as an international language ended. This made it less used, creating a vicious cycle.

Sources:

Ostler, Nicholas. Ad Infinitvum: A Biography of Latin. New York: Walker & Co., 2007.

Erasmus, Desiderius. Ciceronianus; or A dialogue on the Best Style of Speaking. Translated by Izora Scott. New York: Teachers College, 1908.

Avellanus, Arcadius. "Dr. Avellanus' Rejoinder." The Classical Weekly 10: 37-38.

Edit: Fixed spelling, added references, cleaned up formatting.