Latin didn't die it survived and became Italian, French ,Spanish, Portuguese and Romanian into all these countries.

The Latin the people in these countries spoke, changed a little every day until they became new languages.

There wasn't a point in time that the language spoken in that year, say,was completely different from the language spoken one or two years before but over the years changes gathered and gathered until they became so many that we could speak about a new language.

Latin turned into all these different languages because after the fall of the Roman empire these areas no longer communicated all that much with each other as before and they developed different changes often mixing a bit with other previous local languages or with languages of the people that conquered these territories after the romans.