this still isn't making sense to me, is there a simple math example of how this is even possible if encrypt(1) = a, then how can encrypt(1) + 2 ever give me the correct answer?
As the comments in the article state, it's actually all done with operations on polynomials with encrypted coefficients. As + and x can be done on polynomials, it all works out.
The maths behind it is of course much more complicated.
As for a simple maths analogy, consider adding 1 + 2.
I'll encrypt your values by multiplying by 3 mod 7. So encrypt(1) = 3x1 mod 7 = 3 mod 7 and encrypt(2) = 3*2 mod 7 = 6 mod 7. Only the encrypted values are sent, along with the operations I want to perform on the values.
Now this scheme is homomorphic, as I can just add the encrypted values, encrypt(1) + encrypt(2) = 3 + 6 mod 7 = 2 mod 7. That is the only computation that would be done before sending back the result.
To decrypt it, I would have to divide by 2 mod 7 by 3, which gives me 3 (you can easily check that encrypt(3) = 2 mod 7). And indeed 1 + 2 = 3. So the decrypted answer is correct.
Of course this scheme is too easy to crack. The FHE scheme is not.
i think you meant divide by 3 mod 7:
9 / 3 mod 7 = 3