I learnt about payment processors when I tried to understand how some countries adopted crypto currency and use it for payments after they concluded their transactions. At least that is the only reason I know why it exist unless I get proven wrong by someone with more information. Perhaps you can explain more for me.
Assuming you want to buy an item from an online shopping site, the site may not accept bitcoin but allows a payment processor that likes bitcoin on its site. You can pay with bitcoin on the site. The payment processor will accept the coins but pay the site in fiat. You can pay the payment processor by sending the coins on a noncustodial wallet or an exchange.