I received a notice from IRS and it said that I filed my return late. I filed the extension via regular mail, what can I do to prove that I filed the extension?

If you did not use certified or registered mail from the post office when you mailed the paper extension form, then you do not have any proof of mailing.

The IRS should be using the postmark on the envelope for proof of date/time that it was received at the post office.  So without your proof it is just your word against the IRS, which means the IRS wins, sorry.