Server keeps track of which CSRF token is given to which client using cookies (usually some form of SessionID), and stores it on the server somewhere.
It is a very common pattern and all frameworks support it with the concept of "sessions" on the back end.
you don't need to store anything on the server. cookies for that domain are sent with the request and it is enough for the server to check its cookie with the csrf request data.
browsers would send the bank.com cookies with the bank.com request. It is security built into the browser which is why its so important to use secure browsers and secure cookies.
If the malicious user convinces the user to use an insecure browser you can circumvent CSRF, but at that point there are probably other exploits you can do.
You can validate the CSRF is valid by keeping a key on your server and matching that the token you get can be derived from that key.
See Django's implementation of CSRF for more details. CSRF tokens are separate from session and no CSRF information needs to be stored in database to validate CSRF.
depending on why you'are asking the question, * because it decrypts correctly * because it contains some user identifier
People don't usually store sessions in cookies because cookies can't be very big, and session do become big. So what people do instead they store cookies in databases, and put session identifiers into cookies.
CSRF token can be entirely separate from sessions.