They spoke French because they were Normans and married mostly with the continental aristocracy.

Their claim to the French throne was based on the rules of succession and an argument over them (arising from those intertwinned lineages).

An interesting fact is that King Richard I (Sean Connery in Robin Hood with Kevin Costner)'s mother was from South West France and he grew up there, and so he spoke French and Occitan but not really English.

[1]

Henry IV was the first English king to speak english natively since before the conquest. Mel Gibson's Braveheart is particularly ridiculous because everyone would have been speaking various amounts of Norman French to each other in reality. Wallace, the Bruce, Edward I and II, the various other nobles.

[1] https://en.wikipedia.org/wiki/English_claims_to_the_French_t...