I've been wondering why universities require Computer Science students to take a lot of mathematics classes. What sort of things will higher mathematics such as trigonometry, calculus, etc. help me achieve with computer science? I'm especially interested in programming.

Well, from my perspective of 20 years which was mostly outside Game development:

• Computer programs are Logic constructs which is actually Maths. In fact, having gone to Uni into Physics, my very first Programming course was showing us how it could be done entirely as a Maths construct.
• Trignometry is essential in Games programming as a lot of what ends up as code are 2D and 3D calculations.
• Probability, Statistics and Queue Theory are essential in figuring out things like how much load peak load can be on a system or things like the likelyhood of system failure given the likelyhood of failure of each redundant sub-system. By analysing these things one can make appropriate choices when designing a system in terms of how much error margin to add or how much effort to devote to dealing with recovering from certain kinds of failures.
• A number of other Maths elements go into things like Neural Networks or Genetic Algortithms. Other complex AI techniques might involve Monte Carlo Simulations, which is a Numerical Analysis thing.
• Further Numerical Analysis might be used in things like finding formulas that fit desired points so that you create a continous curve that matches a set of desired waypoints. Or at least in understanding how Bezier curves work.
• Things like derivatives and integrals are used in simulating physical processes (so, in Game making also) as well as things like understanding how a discontinuity in the first order derivative on a surface curve would cause artifacts in specular illumination (usefull if for example you're doing mesh generation).
• If you're doing games which are simulations you probably want to know about positive and negative feedback loops and maybe some of the basic stuff that relates to Economics. The maths that relates to Economics is also used in things like multiplayer games which have some kind of Economy.
• For certain kinds of mesh/texture generation it might help to at least know what Chaos Theory is.
• Beyond that, if you're doing Tech in business domains other than Games, certain disciplines often turn out to be useful in understanding the Business and making software for it (just like Trignometry and Calculus, for example, are very important in most Game development but not at all in most other programming areas).

More in general, to me personally it always worked well to be able to understand WHY and HOW things works rather than merely use other people's code, and for that it helps to have a deep groundwork of understanding in, amongst other things, Maths.

An observation I can add to what Aceticon has said is this: 
Higher level maths allow you to execute more effective code with less computing power.
Take the following example for instance, I want to know how much money I will have at the end of the year if I save $100 every month.  With the lowest level maths skill I can write the following code

JanuarySavings + FebruarySavings + MarchSavings + AprilSavings + MaySavings + JuneSavings + JulySavings + AugustSavings + SeptemberSavings + OctoberSavings + NovemberSavings + DecemberSavings = YearsSavings;

 With a just a touch more advanced math I could write

MonthlySavings * 12 = YearsSavings;

The illustration here being that multiplication is just a tad bit more abstract a concept as opposed to addition.  Now take into consideration all the other primitive operations you learned in highschool and apply them to your logical program and the speed and execution of your program will improve.

Advanced math is taking this same illustration to the next degree.  The more tools you have in your Mathematical Tool box the more effective and complex your programs can become.

6 hours ago, RidiculousName said:

I've been wondering why universities require Computer Science students to take a lot of mathematics classes. What sort of things will higher mathematics such as trigonometry, calculus, etc. help me achieve with computer science? I'm especially interested in programming.

As I'm sure you'll discover, trigonometry and calculus are not exactly higher mathematics.

Those requirements are usually come from a notion that studying that kind of thing will prepare you for the other, computer science-y math that you'll have to deal with. There's a certain assumption that going into your program, you're familiar with the notations that you would learn in those subjects, eg. sigma notation for summations. They expect you to have some amount of basic knowledge going in so they don't have to drag everyone up to standard in the first year.

A lot of the "theoretical" computer science classes will essentially be discrete mathematics courses. Those are the courses where you'll learn things like basic set theory, graphs, propositional logic, and complexity theory. In addition to that, most universities require CS students to take some amount of "actual" math and statistics, of course, though typically less so than physics or engineering students might and with more choice in what specific classes to take. For instance, differential equations were not required by my program - I took linear programming and number theory instead. At the time, I was a lot more interested in systems programming, programming language theory, and compiler construction than I was anything related to the stuff the engineering faculties were doing. Looking back, I wish I had taken more abstract math courses. Abstract algebra (group theory, rings, etc.) would have been more interesting and applicable than linear programming has turned out to be, and maybe then I would actually understand that "monads" thing in Haskell.

More "practical" CS courses won't be especially math heavy - my software engineering classes barely even required arithmetic! - but you'll still use the stuff you learned in the theoretical courses. Whether those math fields are useful in your professional life will be highly dependent on what kind of work you're doing. If you're doing graphics or gameplay programming, trigonometry and linear algebra will be indispensable. If you're doing physics, calculus will be very useful. If you're doing lower-level systems development, trigonometry won't be terribly helpful (when's the last time anyone here needed to know the arc-cosine of some number when implementing a memory allocator?), but complexity theory and number theory definitely will be.