Re: there's more to understanding this than just reading the article
Thanks for the wiki link. Forgive me if I am wrong, but the idea seems simple then.
A "space-time crystal" is a theoretical atomic or other particle system that returns to a previous state, thus breaking time symmetry. Time symmetry involves being able to run a system backwards and forwards and getting the same results (but in reverse and visa versa).
A "time crystal" would break this because when you set it up, it now "forever" runs like in a loop. So if you try to reverse it back to before you started the loop, you instead end up going in a loop again, but this time "backwards" in time.
It is effectively the hope of setting up a loop of dominoes that knock themselves over again and again. Or perhaps like sticking both ends of a VHS tape and taping it into a loop so it plays forever?
I would assume that to get something like a time loop, you would have to forgo any loops in space (space-time equivalence). Otherwise the entire idea is impossible.
TL:DR you can have a roundabout in a playground that returns to the start position every rotation, but over time looses energy. OR we can have a roundabout that moves through time never loosing energy, but it can never move.
These people want both and in effect, a Magic Roundabout. ;)