What do faculty members mean when they say "happy to find a time some time to chat more about it..."? Sometimes in correspondence with faculty I get emails that end with a sentence like "happy to find a time some time to chat more about it though, just let me know."

What medium do people usually mean when they say this? Is this an invitation to schedule a phone call? Or because this is on an email is it best to stick to to sending responding with a detailed email about the said topic?
The "just let me know part" invites you to let them know your preferred times and propose a way to have a chat.

The typical medium to let them know would be via email. And then the typical medium to chat would be in person. But that's it's not so important. You've been invited to make a suggestion without much advice. So any response that is both reasonable and polite should work. Even if the professor will "counter" your suggestion with an alternative.