Omitting "by" preposition and the resulted phrase
I paid for it by using my credit card. - "by using my credit card" is a prepositional phrase consisting of a preposition + a gerund phrase. It is a bound modifier and modifies "paid"
I paid for it using my credit card. - "using my credit card" is a participle phrase used as a free modifier, i.e. it modifies the main clause "I paid for it" adverbially.
This latter point is probably clearer as Using my credit card, I paid for it, which is to be understood as "I was using my credit card when I paid for it."