I've always known that higher VRAM will help at higher resolutions, but why? What exactly does it store? I understand that when you run out of space on VRAM, you have to load it directly off the hard drive which causes FPS drops, but what exactly is it loading off the drive?
VRAM is just like normal system RAM except it is used solely by the GPU. It stores whatever information that the GPU currently needs access to so that it can access that information relatively quickly (much faster than waiting on a hard drive or even waiting for system RAM).
As for what exactly is stored in VRAM, nowadays the vast majority of what you'll find in VRAM is the data the GPU needs to perform the calculations to turn 3D geometry data (the input) into a 2D projection (the output - or what you see on your monitor). There are some other things that need to go into VRAM such as frame buffers, which temporarily store the images that need to be displayed on the monitor (if you've heard of double-buffering or triple-buffering - these frame buffers are what those terms are referring to). However these extra little bits use an insignificant amount of VRAM compared to the total available space.
More VRAM helps at higher resolutions because the higher the resolution the GPU needs to render, the more data it needs to work on its calculations. That data needs to be stored in VRAM for efficient access so more VRAM = better performance. However there is a caveat to this in that having more VRAM than the GPU needs will not further increase performance; if the GPU only needs to load 2 GB into VRAM, having for example 3 GB VRAM will net you nothing over having 2.1 GB VRAM.