Asymmetric cryptography means that the private key doesn't need to be stored on the client or the server.
If you have someone's public key, and no other key material whatsoever, you can encrypt something in a way that only they can read it. Even you can't decrypt it.
(Because asymmetric cryptography is slow, if your messages are large you'd typically encrypt the data with a symmetric key, and then encrypt just that key asymmetrically.)
I've PR'd this project into the restic/others list.