Cross-Site-Request-Forgery, CSRF, isn't an attack directed to cookie-implementing sites, it's an attack that involves the ability to cause a user's browser visiting site A to invoke an action on site B - normally one in which they have to log in to access, but it isn't necessary.
For instance, assume you have a simple "Contact Us" form on Site B. This form is publicly accessible and requires no user-login. If Site A can submit this form from a client's browser via javascript (or Flash, etc.) - then this would be considered CSRF as the "Contact Us" form will appear to have originated from the end-user who never actually visited Site B.
Now, this attack is far more dangerous when actions are more complex than a simple "Contact Us" form, say, "Transfer money from account X to account Y". These actions generally require a user to be logged in to the site, which also normally use a cookie of some form (Session IDs are sent back and forth from the browser to the server as cookies). Without CSRF protection (such as tokens), the "Transfer" action could be performed as long as the user has an actively opened session. If, however, the site actually saves a cookie to allow a "Remember Me" function where the user doesn't need to submit their credentials each time, the CSRF should be able to submit without the user having an active session as well.