A proxy re-encryption is generally used when one party, say Bob, wants to reveal the contents of messages sent to him and encrypted with his public key to a third party, Charlie, without revealing his private key to Charlie.
[1] Bob could designate a proxy to re-encrypt one of his messages that is to be sent to Charlie.
Now if Bob sends Charlie a message that was encrypted under Bob's key, the proxy will alter the message, allowing Charlie to decrypt it.
This method allows for a number of applications such as e-mail forwarding, law-enforcement monitoring, and content distribution.
A weaker re-encryption scheme is one in which the proxy possesses both parties' keys simultaneously.