This page is really easy to understand, there are only 4 fields:
New First Name:
New Last Name:
If you don't know where to look for the player ID, just click the link of the player you want to rename. In the address bar of the web page that comes up, you'll see a URL that looks something like this:
Just copy/paste the player id from the URL into the text box. Bringing up the player card for the player to be renamed is a best practice anyway, so you can refresh the page and verify that the rename occurred as expected. The error handling on this page does not work properly, so if you make a mistake typing your password, it will say the rename was successful, even though it was not.
Anyway, the Player ID field should contain the ID of the player you want to rename. The player ID must belong to a player in the league that you are in context of, otherwise the rename will not be successful.
New First Name should contain, yes, the new first name of the player.
New Last Name? You guessed it, the new last name of the player.
Your password should contain your CSFBL game password, not your forum password.
Once all fields are entered, click the Submit button. If you followed my advice and kept the player card page open, refresh the page and view your new named player. Use special characters with caution (&, %, @, etc) as these may or may not cause problems with the page. I've not tried it, but these characters commonly cause issues in web sites.