This is a response when someone asks you a favor or just being polite when asked to do something. The English translation is a bit misleading in this case and more literal because "szívesen" means "gladly" or "you are welcome". A common way you would reply in English is "Yes, of course" or "Yes, no problem".
I thought about examples and could barely come up with anything because the phrase itself is not that common. We do use "szívesen" a lot but that mostly in response to a thank you ("köszönöm") and usually not before that. It makes sense if the person asking is unusually polite or very desperate.
Szív - heart; szíves - with heart/hearty; szívesen is the adverb. "Heartily" might be a weird translation for it, but yours is just as close.
If you're any familiar with German, you probably stumbled across the word "herzlich" for things you put some love in. That has the same meaning.
Ah, I mixed it up a little. Tessék is more like "here you go" in English. Kérem would be closer to "please".
Correct me if I'm wrong, but I'm pretty sure that when you're getting or buying something from someone, you'll use a dialogue like this:
- Kérem azt add.
- Tessék. [adja]
- Köszönöm szépen.
Right? The same dialogue in English would look like this:
- Hand me that, please.
- Here you go. [gives it]
- Thank you (very much).
- You're welcome.
If "Szívesen" is an answer to "Köszönöm", then it can translate as "You're welcome", since that is the answer to "Thank you." If not, how would you use szívesen?