This licence is referred to along with several paragraph of comments in a Java API based project I'm using for research.

I'm having difficulty understanding some of the law-speak in the license. Especially this part:

You must retain, in the Source form of any Derivative Works that You distribute, all copyright, patent, trademark, and attribution notices from the Source form of the Work, excluding those notices that do not pertain to any part of the Derivative Works; and

What I'm doing is creating my own classes that call the methods in the provided work, but not modifying them directly. Does this mean my file should still contain their copyright notice on the top?

Generally you don't need to do so, assuming you are not modifying Apache code itself, but are simply writing new classes of your own. You might want to insert your own copyright notice however, if you plan on distributing your code to others in a source format.

Hmm well, let me explain like this:

The package contains this:
Class A
Class B
Class C

All of the above contain copyrights.
What I'm doing is creating a with code such as this:

ClassA a = new ClassA();

So do you think I need to insert that copyright disclaimer in

In my opinion (but I'm not a lawyer), if you're not modifying the original code then there's no problem. If you plan to distribute the original methods as part of your product then you simply need to retain all the original documentation (licences etc) that came with them.
You certainly should have a copyright statement in the code you wrote, and consider using a common licence (such as the GNU GPL) to protect your own code.
(but I'm not a lawyer)