It doesn't, but it does help us with identity. We don't know that the message was actually from the real Snowden, but we do know that it was from someone possessing the same private key as the previous account that we assumed was Snowden.

The possibilities in descending likelihood are: 
 - this is Snowden
 - the previous was not Snowden
 - the previous was Snowden and someone has stolen his private key

Most of the time that last one has a higher probability, but ... Snowden. 

What makes public key cryptography magic here is that we don't have to trust an institution's word that the account hasn't changed hands. As long as the private key is secured, it isn't possible for anyone else to pretend to be someone.