You might be thinking, "But what about the next message from Jim?" but that message is encrypted with a new key so the previous key isn't useful, your Signal works out what that key will be and remembers it until it receives a message from Jim.
It's a ratchet, you can go forwards but you can't go backwards, if I didn't keep the message Jim sent me last week then even though you've got the encrypted message, and I've still got a working key to receive new messages, we can't work back to decrypt the old message.
You might also be thinking, "There must be a long term identity key so that I can tell Jim and Steve apart?". Indeed there is. But Signal doesn't use this to sign messages since that's a huge security mistake, instead this long term identity key is used to sign a part of the initial keys other parties will use to communicate with you.
This design deliberately means you can't prove to anybody else, who sent you anything or what they sent. Sure, you can tell people. You can dish the dirt to your spouse, your friends, the Secret Police, but you can't prove any of it cryptographically.
