They charge $59 to file Fed taxes plus a processing fee if they take the charge out of your Fed return - how much is the processing fee?

$40 and $45 in  CA.

You will pay a refund processing fee of $40 ($45 in CA) to have the third party bank receive your refund from the IRS and take out the amount you owe to TurboTax----you can avoid that extra fee if you pay the fees with a credit card or debit card BEFORE you e-file.

REMOVE REFUND PROCESSING FEE

https://ttlc.intuit.com/questions/1900517-how-do-i-remove-refund-processing-fees-in-turbotax-online-and-change-my-payment-method

Be aware that there are other concerns when you use "refund processing."

Please make note that if your refund takes too long to arrive (IRS delays, etc.) then you will also be agreeing that the TurboTax fees can be extracted from your bank account even before your refund is issued by the IRS.  This comes as an unpleasant shock to many people every year.  

 If you can possibly pay your fees upfront using a debit or credit card, it will save you that extra fee and even more importantly, the possible aggravation and delays that are often the result of poor handling/poor customer service by that third party bank.  We see way too many instances where refunds are delayed for months when users choose refund processing. There have been lots of complaints; there are a number of things that can go wrong when you use refund processing.  I suggest avoiding that situation if at all possible.