Explanation follows (skip to the third paragraph if you want to know the beer analogy)
When you do an LLR test you start with some small initial number then repeatedly square it and subtract 2 for (n-2) iterations (where n is the n in k*2^n - 1) - this is the iteration counter you see when you running a test on PRPNet.
Clearly the numbers involved get pretty big (even though you do all the arithmetic modulo the prime number you are testing). So, when you get to the end of the iterations, you look at the number you have left - if it is zero, then you have found a prime, if not, then you have a composite, and a large non-zero number. What the residue is is just the lowest 64 bits of the big number - enough that if two computers do the test and come up with the same residue then we can be sure they have done a correct calculation to very high level of confidence.
In terms of beer, you could think of the residue as being a small sample taken from a large tank at the end of the brewing process - it would be impractical to sample the entire tank before bottling and distributing it. Just by looking at a small sample, we can be fairly sure about the rest...