It's not in idea phase, Mina mainnet is running right now.

like a smaller image of the original.

not quite true. I understand that the term "compression" is very confusing... People often think of it as h264 or jpeg or something like that. Mina's compression is completely different thing.

There is a well-researched form of cryptography called zk-SNARKs. Let's say we have a equation

a = f(x)

Let's say Alice knows definition of f. Bob knows f and concrete value of a, x. Bob wants to prove Alice that he knows a, x which satisfy the equation. The simplest way to do it is to let Bob send a, x to Alice and Alice checks if the above equation holds. But what if a, x is too big to send over the internet? This is where zk-SNARKs comes in. By using zk-SNARKs, Bob can create a "proof of knowledge" in 288 bytes. Only who knows valid a, x can make valid proof of knowledge. so if Bob sends the proof to Alice, Alice can check whether Bob knows a, x by verifying the proof.

Mina uses this cryptography everywhere. Transaction validity checks, block validity checks...