I want to take this class because the professor comes highly recommended, but I'm a little worried that I won't be entirely prepared for it. Normally this class requires Real Analysis as a prerequisite, and even though the professor explicitly states that Analysis isn't required, I fear that not having that background/mathematical maturity will hold me back.

I have taken three semesters of calculus and a course on differential equations and linear algebra. These are all of the prerequisites... but if I were to prepare for this course, where would be a good place to start?

I took a class like that. The only thing we used from real analysis was the implicit function theorem, so it's good to have seen that. But actually, the class itself is where I got a lot more intuition about the implicit function theorem because one of the most natural contexts where it comes up is to prove that some level surfaces of a function on R^3 are "regular surfaces". So, if you're a good student, I think you could handle it already if you really know your stuff from calc 3 and linear algebra.

It wouldn't hurt to look into real analysis, but as far as the implicit function theorem goes, I think the geometry class would be more helpful for understanding it than the other way around, if it's anything like my experience. There are a lot of real analysis books. Not sure what your style is. My favorite is A Radical Approach to Real Analysis for its historical motivation (take the history with a grain of salt, though), but I haven't read that many intro to real analysis books. From what I've heard, I would probably like Understanding Analysis by Abbott.