Similarly - it would not be natural modern English but you'd be understood perfectly well.
I would guess that "with pleasure" or "(it's) my pleasure" are the best idiomatic equivalents - both give just a little more formality in English without going over the top. Maybe "with great pleasure" if you're really piling it on. In my small business I say this to my customers all the time, as well as "you're very welcome" or "you're most welcome".
But then I'm a complete newbie in Hungarian so it's quite possible I'm getting a nuance wrong from the other direction. ☺
Going with https://wikiszotar.hu/ertelmezo-szotar/Sz%C3%ADvesen , szívesen can mean readily, with pleasure, kindly. So an alternate (imperfect) translatation could be "It was my great pleasure to help you".
As for etimology, szívesen is an adverb, generated from szíves. Szíves is "kind", as in "szíves fogadtatás" - "kind reception" (but not the "someone is being kind", we use a different word for that "kedves"). And if we really want to go deep, then szíves comes from "szív", which is "heart". Hope that will help.