And what happens when you follow your providers server instruction and verify your account at their webmail page?

What is your question? The message you have quoted is straightforward: it wants you to log in to your email via the internet in order to verify the account. The log in procedure is described here: http://www.sbcglobalnetemail.org/www-sbcglobal-net-login-to-your-sbcglobal-att-email-account-online/ There's a link on that page for 'email settings for sbcglobal.net' which tells you the settings you need when you set up the account in Thunderbird. If you do that correctly, and when you have verified the account, it should work OK.

Did that. Thunderbird still won't let me receive email.

if it has that error, the issue is not with Thunderbird but your mail provider... I suggest you waste some time talking to ATT and Yahoo. You might get something sensible from them.