I am reading a paper that I barely understand. How can I make the best of my mentor in this situation? Half of my inability to understand the paper comes from not being acquainted with the folklore of this particular part of the field, and the other half might be because I am lacking general background knowledge. I cannot just go to my mentor asking them to fill in every single step I am missing. I am also sure they can put things in perspective, or just say something illuminating.

What kinds of questions are reasonable in this case?
Jumping in to a new sub field can be hard. The vocabulary is new and you don't yet have the insights of those with more experience. 

If you trust your advisor, you could just say what you say here, but ask for the important background papers (or books) that will help bring you up to speed quickly. This is what a colleague would do, so it isn't impossible for a student as well. 

As Patricia Shanahan suggests in the the comment, a typical way is to go back through a chain of references until you come to things you understand and then work forward again. This is hard and time consuming of course. It also implies access to a good library (and a good librarian). 

But one way to attack the problem is to write down your questions about the paper as they arise. I'd suggest using index cards for this. One question per card. This leaves space for you to supply answers (or references to answers) as you come across them and your understanding improves. 

Those cards, once you whittle them down to a reasonable number, can be used both for research (searches, say) and for a conversation with your advisor. Getting answers to a few key questions might let you fill in the rest. 

Everyone starts out knowing nothing. In math, insight into some things may help give insight into others, but it isn't guaranteed.