"Szívesen/You are welcome" is the answer if someone says "Köszönöm/Thank you". In this case: "You are welcome in my house", you are welcome means "Üdvözöllek".
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.
I believe that "szív" means heart, so would one meaning of "szívesen" be "with all my heart", which equates to "very gladly"?
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.
How would you use this sentences in a context? Is it like a "yes, thank you" or what?
It's definitely not "yes, thank you". "yes" is very confusing there and it's hard to come up with a conversation with it in this sense. I can think of one but "szívesen" is used in a different sense there so I'd rather not confuse you.
Would you reply with that to a request? Like "could you pass me the salt?" - "Yes, gladly" - As one of the translations offered for szívesen is "gladly".
No. That is curt and most impolite. Are you grumbling angrily? Maybe then, but I wouldn't.