Linguist here. There is no why exactly, but we can describe the process.

For the sake of the uninitiated: in Latin, nouns and verbs conveyed their part of speech by an ending attached to a stem. In Modern English, if you wrote "am hungry," the I subject is implicit in am. So it was for all verbs in Latin in both singular and plural. Likewise with nouns, we designate part of speech by position ("I am sending it to him" where it is the direct object and him is the indirect object), Latin does this by noun ending. Because of the endings on the words, Latin has mostly free word order. "I am not him" vs. "him I am not" vs. "am him I not" vs. "not I am him." You can see that these sentences are all understandable (if not weird) to the modern ear.

So in popular Latin (i.e. Vulgar Latin), it was typical to "clip" the ending of words off. The most obvious example in English is "g-dropping," so you'll hear Southern Americans talkin' about runnin' and jumpin' and skippin' and lovin' and swimmin', and so on (incidentally, Shakespeare dropped his g's too). Here's a video that describes the process a little bit.

So, without clearly hearing the noun and verb endings, speakers of Vulgar Latin opted to use word order to convey meaning. You have to remember, though, that these are natural processes and real people. They didn't wake up one day and decide to stop using SOV and instead use SVO. It just happened naturally.

As for when, that's not easy to tie down. It was likely happening around the 2nd to 5th centuries, but we have no spoken records and can only go by written records (on the sometimes-tenuous assumption that written records reflect spoken forms), and written records for Vulgar Latin are scant compared to the upper class Classical Latin which preserved the SOV longer in written forms.