How does the question from payments with 2024 federal extension work? Does this mean if I filed an extension and then paid later I can take credit for it his year?

Unfortunately, there are a lot of credits so it's hard to determine which particular credit you are referring to.  As for the extension, this only provides a longer period to file your tax return but does not extend the time to pay the tax due.  Any tax due needs to be paid by the original due date of your return or you could face late payment penalties and interest.  

Please see what if I can't pay my taxes for more information.