This is fascinating, and seems like a Big Deal. A bit beyond my skill level with math though. There's a very good comment by IBM after the article that attempts to explain how it actually works. They offer this TLDR at the end:

TLDR Huge polynomials are what are operated on instead of plaintext values by hiding the real data in the polynomials with intentional noise to make it infeasible to extract the real data without the decryption key that knows how to remove it. All operators are bootstrapped from addition and multiplication and elegant modulus trickery. Noise is managed at each operation because the noise compounds with every logic operation you do. The result of an operation chain or circuit be it one operation or infinitely many, therefore, has some noise but can be decrypted by the person with the encryption key. The decryption effectively drops all the noise and hones in on the bits that matter. The person doing the computation is doing a lot of adds and multiplies on big polynomials so they cant understand what you are really trying to do.

As someone that's been on the lookout for research into FHE, I find this fascinating! I've been trying to read through their source code to figure out how the encryption scheme works, but all I've learned so far is matricies and tensors. I guess it's time to learn about tensors and what they are.