In the books...
When he was young Rhaegar was a bookish bloke, until he read the prophecy of "The Prince That Was Promised" and came to believe it was about him so he picked up a sword to become a great warrior. Later, he came to believe that it wasn't actually about him, but his children, and that "The Prince" wasn't one person, but three because "the dragon has three heads". At his point he had two children, and his wife was unable to bear more.
More specifically why he chose HER, there was a great tournament in the Riverlands where Lyanna took umbrage at the poor treatment of Howland Reed (The guy who stabbed Arthur Dayne in the neck during his fight with Ned) by some southern Knights and (we can gather from Meera's telling of the story) armored herself in the guise of a mystery knight (which Rhaegar himself was rumored to do in order to assure no one went easy on him) and defeated several of them. That same tournament was where Rhaegar crowned Lyanna with blue winter roses (a recurring symbol linking her to Jon in visions and prophesies) the "Queen of love and beauty" over his wife, the only available recognition of the victories which she could not claim, due to being neither a man nor a knight.
From that point onward it is safe to assume a torrid love affair ensued, with Lyanna eventually running from her promised marriage to Robert. This set in motion the events that lead to Ned's brother and father being killed by the mad king and eventually the war that put Robert on the throne.
From that point onward it is safe to assume a torrid love affair ensued, with Lyanna eventually running from her promised marriage to Robert. This set in motion the events that lead to Ned's brother and father being killed by the mad king and eventually the war that put Robert on the throne.