Robert Falconbridge (Robert Falconbridge, Jr., to be precise) only appears in one scene of the play: Act I, Scene 1, where he comes before King John to challenge his older brother Philip's claim to inherit their father's land. Robert Falconbridge claims that he and Philip the Bastard don't share a father at all. He says that the Bastard's true father was Richard the Lionheart, the previous king of England, who slept with their mother while Robert Falconbridge, Sr. was away.
Even though Robert Falconbridge, Jr. makes a convincing case, King John sides with his (half-)brother because of a legal technicality. In the end, however, Robert Falconbridge ends up getting what he wanted, because Philip gives up his claim to the land and chooses instead to be openly known as the son of Richard the Lionheart. At this point, Robert Falconbridge, Jr. disappears from the play entirely. Presumably, he's enjoying his father's land in peace.