If you rephrase this question to ask about "the influence of Norman French on English", you will discover a myriad of searchable sources and references, and the book Contact: The Interaction of Closely Related Linguistic Varieties and the History of English (2016 Edinburgh University Press).
The linguistic influence comes from the societal change: the French-speaking invaders established their government and conducted their affairs in Norman French. People who had to deal with their new lords learned their language; then vocabulary diffused through the rest of the society. When the Normans had a concept that Middle English did not, the French word became dominant: parliament, joust, mustard. In other cases it became fashionable to use a French term to distinguish oneself from the masses, and both words remain with similar meanings: cow/beef, sheep/mutton, pig/pork are especially obvious. Seek a social change to explain a linguistic change and you will generally be successful.