There wasn’t a moment in which the people stopped using latin, it was more a process taking some time. Already during Roman times, the spoken Latin was influenced by the substrata of the languages spoken before. The first written things that testify that spoken Latin was becoming other languages in Italy are from the end of the VIII century and the beginning of the IX, in particular the indovinello veronese is considered the first written document to show this gradual change.

As for why, all languages evolve. Already during Roman times, there were different words used in different areas and there were words used in the written Latin and other versions used in spoken Latin (ex. equus vs caballus). Without the unifying force of the Roman Empire, the spoken Latin started to evolve in different paths, based on the substrata (the languages spoken before Latin) or the influence of the languages spoken by the conquerors (ex. words taken from Germanic languages)