Paranoia about untrusted hardware is absolutely warranted but just to try and convince you of what this is trying to do. Imagine you have a file that you have never shown anyone, and some untrusted host wants to convince you they also have that file. They can prove this to you without you revealing the file to them (or them to you) by having them send you the hash of the file that you can compare to your own. If it matches there is an overwhelming probability they also have the file even though you completely distrust them or their execution environment. In other words, you are able to verify the computation (via the hash) was executed even though you don't trust them. Cairo is using similar techniques that let you verify other forms of computation than just hashes using more advanced primitives.

The obvious caveat to this just as with hashes is you trust the underlying cryptography is secure.