Leo opens doors—you know, for a living. But he also does this metaphorically for the other characters in the novel. Most obviously, his book The History of Love is responsible for bringing together two of the love stories in the novel: Alma's parents and the Litvinoffs. (Feel free to read a sexual metaphor into the connections between key and lock as well. We'll leave that one to you, though.)
The book Leo wrote also indirectly brings him into contact with the outside world, most notably with his son and with Alma Singer. So, if Leo is a locksmith, and the book allows him to open doors, then The History of Love is the literal key to The History of Love.