What should I focus on before starting a research internship?
You should always make sure your basic understanding is at a very good level in any research project, particularly in the sciences. An important part of science and mathematics is being able to communicate your ideas in very simple terms. I think Einstein said that "if you cannot explain it simply, you do not understand it well enough." If all you can do is prove things but not explain what the results really mean or how they connect to the basic components of your field, then you're not really doing science or math - you're only shuffling around symbols on a page. It's a bit like a computer: computers can prove things about math (automatic theorem provers), but would you say they're doing math if they don't understand what it is that they are doing? All they do is shuffle symbols around and come to conclusions based on basic axioms inputted by the user.

This isn't to say that you shouldn't pick away at the problem immediately. You should maybe poke around and get an idea of how things go but if you are put on the spot and asked a basic question and you can't answer it, you'll look like you haven't done your due diligence. You're always going to have some gap in your understanding, even at the fundamental levels, but that doesn't excuse not having any understanding of the basics.

Here's another take: if you end up solving the problem you're given, your professor might ask you to write up a paper on the work to be submitted to Journal XYZ. If all you can do is explain the steps you took but not connect it to the things the reader knows, your paper will be completely dead in the water and will likely get shrugged off. A good paper has good results. A great paper has a very good narrative with good results that is very clear and understandable. Even if you don't write the introduction yourself, throughout the paper you will naturally connect results or ideas back to things others have done ("Our approach relates to this work by [authors] in such and such ways," or "Employing a common technique, we have..."). If you don't have a basic understanding of the things in your field, you can't make this connection for the reader (which forces them to hopefully come to this realization themself). This is not a skill you should expect to have immediately; it takes time to learn and hone this skill but it is an important aspect of paper writing.

A personal anecdote/cautionary tale: I started on my current project a few years ago in undergrad. At the time, I had a very rough understanding of advanced mathematics. I knew enough measure theory and functional analysis to vaguely understand ideas and basic proofs but not enough to know what are acceptable procedures and what are not. I neglected the basics - not because of my own doing, but lack of proper supervision - and ended up too focused on my project. It's a bit like putting blinders up: you won't have any understanding outside of your project and can't properly judge the validity of what you're doing. I was rearranging integrals like it was going out of style because I didn't know that I couldn't do that in my situation due to skipping the basics. I ended up with results but they were not acceptable proofs by any means. Since I was so focused on my project, I ended up not having much of an understanding of the basics and looked like a deer in headlights when people asked me about the basic, very technical details of the mathematics I should have been using. Since then, I've more than made up for all of those deficiencies but had I not unintentionally neglected the basics, I wouldn't have wasted so much time (read: years) doing stupid things that were not valid at all.