Correction: session keys secret and symmetric not public asymmetric
"The feature, a type of key-establishment protocol known as forward secrecy, ensures that each online session is encrypted with a different public key and that corresponding private keys are never kept in long-term storage."
The purpose of the Diffie Hellman protocol is to establish a secret symmetric session key between 2 ends of a communication link, lets call them Alice and Bob. Alice and Bob still need to sign something as part of the session initiation stage with their own asymmetric secret keys which the other can decrypt with the corresponding public keys so Alice can verify Bob's identity and Bob can identify Alice's, which defeats a man in the middle attack.
Once a DH symmetric key and endpoint identity has been established , the symmetric session key can be used to encrypt and decrypt everything for the duration of the session and can then be securely deleted at both ends.
Knowledge of an asymmetric secret key after the session is of no use in recovering the deleted session key so previous recorded sessions are still protected. This knowledge could be used to compromise the owner of the compromised asymmetric private key in future sessions by impersonation and MITM attack, with Eve creating simultaneous sessions to Alice and Bob and knowing both DH session keys, pretending to Alice she is Bob and pretending to Bob she is Alice. But if both authenticate each other prior to communication occurring, Eve would need to know the secret asymmetric keys of both Alice and Bob.