Read the full text of Henry VI Part 1 Act 2 Scene 5 with a side-by-side translation HERE.
We meet Mortimer. He's dying, unable to walk, and in prison; he even says he has "desire to get a grave" (2.5.15). He does want to know if his nephew is coming, though.
The Gaoler says that his nephew will come, revealing to the audience that Mortimer's nephew is none other than Richard. Guess he's finished dinner.
Mortimer laments that he used to be a great warrior, but since Henry of Monmouth (Henry V) began to reign, he's been locked up. He says Henry's reign has been bad for Richard, too. He praises death again and wishes his nephew well.
Richard arrives and they greet each other warmly, then Richard describes the quarrel he just had. He also asks Mortimer why his father was executed for treason. Turns out he doesn't know.
Actually, it's the same reason Mortimer is in prison, and he says he'll tell Richard as long as he doesn't die before he finishes the story. Talk about heightening the suspense…
Mortimer tells a long story. Basically, he says he had a better claim to the throne than Henry IV, the current King Henry VI's grandfather. When a group of noblemen tried to put Mortimer on the throne, though, they were killed and he was imprisoned.
Then Henry V inherited the throne from Henry IV (original names, huh?), and Richard's father, the Earl of Cambridge, tried to put Mortimer on the throne again. And again, it didn't work—Richard's father was beheaded, and Mortimer is still imprisoned.
Richard and Mortimer now move on to the present. Mortimer has no son, so as his nephew, Richard is his heir. Basically, this means Richard could be king, at least if he and Mortimer are right about the succession.
Mortimer warns him to be careful, especially since the house of Lancaster has been so successful.
Mortimer tells his nephew not to be too sad, but to plan Mortimer's funeral. Then he wishes Richard well and dies. Talk about an intense scene.
Richard wishes his uncle's soul well and says he'll give him a good funeral. Richard also seems to be thinking about his uncle's advice—he isn't telling anyone yet, but it sounds like he might be interested in the throne ("Well, I will lock his counsel in my breast,/And what I do imagine—let that rest" (2.5.119)).
Richard heads off to Parliament, hoping to be restored to the title of Duke of York held by his family before his father's execution.