I had the same "problem" as you. What finally made me feel I sort of cracked it was those videos. The way I think of it now is: They let you do matrix multiplication. The internal state of the computer is the matrix, and the input is a vector, where each element is represented by a qubit. The elements can have any value 0 to 1, but in the output vector of the multiplication, they are collapsed into 0 or 1. You then run it many times to get statistical data on the output to be able to pinpoint the output values more closely.
I assume there's probably many more complex computational problems outside of my domain that QC can help with. Does anybody know of any?
More precisely, given f: 2^n -> {0,1} which is guaranteed to hit 1 exactly once, Grover finds the one input which hits 1, and it does so using about 2^{n/2} queries of f; but the constants happen to line up so that when n=2, exactly one query is required.