I'm thinking specifically of Anne and Mary Boleyn since it was googling the history behind 'The Other Boleyn Girl' that made me wonder, but when I was doing the research I came across a few sources that mentioned that this was relatively common. I know fostering was a thing among the nobility, but why would this happen between two rival countries who regularly go to war?
England was conquered by French-speaking nobility in the 1000s, the Norman Conquest. For the next several centuries, the English and French noble classes remained strongly tied together, with frequent intermarriages between them. Eleanor of Aquitaine, for example, was the largest landowner in France and was courted by and eventually married first the King of France and later the King of England. Her sons were Kings of England and inherited her land. The desire of the French to gain control over this territory was a big part of what started the rivalry between the two nations. This eventually led to the 100 Year's War (1300s-1400s) between the two, which is sort of the kickoff of their long history of fighting each other. By the end, England had lost control of pretty much all the territory they'd had on mainland France.
Despite the wars, however, there were still close cultural connections. And to a certain extent even because of the wars, a certain percentage of English nobles spent time at the French court ---- get in trouble in England and the King's mad at you? Maybe you're a rebellious younger brother or cousin thinking of making a play for the throne? Guess who's happy to play host to anyone the King of England hates --- the King of France.