It's simple conceptually: you find an encryption method Enc that guarantees `Sum(Enc(x), Enc(y)) = Enc(Sum(x, y))`. That's ultimately all there is to it. Then, you give the server enc_x and enc_y, the server computes the sum, and returns to you enc_sum. You then decrypt the value you got and that's x+y.

Since lots of functions behave in this way in relation to sums and products, you "just" need to find ones that are hard to reverse so they can be used for encryption as well.

Unfortunately this turns out to not work so simply. In reality, they needed to find different functions FHESum and FHEMultiply, that are actually much harder to compute (1000x more CPU than the equivalent "plaintext" function is a low estimate of the overhead) but that guarantee the above.