It wasn't a conscious change and it didn't happen suddenly. All Languages change over time, and the Latin spoken by people in different areas changed differently. The people of Rome spoke Latin, but as it changed more and more it began to be mutually unintelligible with Classical Latin, transforming into what we know as Italian. The same process happened in Spain, France, Romania, and a few other places.

Another thing to recognize is that our modern perceptions of language don't always fit the situation in the past. The standard Italian language is based on the dialect spoken around Florence, and it was proclaimed the national language in 1861 when Italy was unified. People in other parts of Italy at the time spoke very different dialects and many still do today, and these dialects are not always mutually intelligible. There exists a situation called diglossia, where people speak and understand both their local dialect and the standard language.