If Alice is sending something to Bob, Alice “locks” the box with Bob’s public key. At that point, only Bob’s private key can unlock the box. If Eve comes along, she cannot open the box with Bob’s or Alice’s public keys because they can only work the lock counterclockwise.

To prove to Bob that Alice sent it, all that needs to be done is for Alice to nest 2 boxes, the inner box being locked with Alice’s private key, and the outer box locked with Bob’s public key. Then, when Bob receives the package, he unlocks it with his private key (proving that he was the intended recipient, and that its contents are secret), and then unlocks the inner box with Alice’s public key (proving that Alice was the sender, and the contents were not changed since she sent it).