When did English, an initially Germanic language, become so influenced by Latin? How did it happen?
Not initially Germanic, it is Germanic. It has borrowed tons of Latinate vocabulary but borrowed vocabulary doesn't change a language, just expands it. English still doesn't come from French or Latin by any stretch of the imagination.
There are a few waves of Latinate lexical borrowing:
Pre-English borrowings, words borrowed into Proto-Germanic or West Germanic before English was a distinct language by itself. Words like Latin caupo > *kaufaz > cheap, Latin vinum > *wīną > wine.
Romano-British borrowings, cultural words left behind by the Romans to the Brittonic languages, which were picked up by the invading Angles and Saxons, including place-names.
Religious borrowings, when the Angles and Saxons converted to Christianity.
Norman invasion, brought mostly French words but a quite a few Latin words as well.
Renaissance and Enlightenment eras, the largest concentration of Latinate borrowing in English
More or less, as new periods of language contact were opened up, you can see English and its ancestral line picked up newer words to adapt to the new situations. This is very common among the world's languages.
In 1066, the Normans, who spoke (a version of) French invaded and conquered England. For a long time, French was the dominant language in England, spoken by the nobility and administrators while the original Anglo-Saxon language (also known as Old English) was primarily spoken by the commoners. Over time, the two languages sort of blended together, with the nobility becoming anglicized. However, a lot of French vocabulary entered the English language.
Hi! not discouraging one of the linguistics historians from contributing here, but FYI, you can get started in the FAQ