An access token is essentially a non-fungible token (NFT) which authorizes access to private content for the owner.
Upon proving the ownership of an access token, the address is extracted which then gives cryptographic proof that you own that address and can then access the private content.
An existing real life example could be a ticket to a football game. The ticket you purchased and now own has a seat number and is unique in the sense there is only 1 matching seat and by holding a paper ticket you have proof of ownership for that seat. The paper ticket you hold allows you entry and use of that seat for a set period of time. Access tokens work in a similar way, with the only difference being that ownership is provable on the blockchain.