If I understand the article correctly, it is a less strong constraint on the signal than being periodic, like an oscillator would be. The signal is only periodic for specific value of time, in between those value of time the signal is not periodic.

To give a music analogy, the constraint is like always play the same chord on the first beat of each measure. In between you are free to play whatever chords you like.

How does a physical system can acquire such property? I have no idea that's the mystery a physicist will have to explain.

I think the difference is that an oscillator requires a continual driving force to stay periodic. Stop applying energy and it winds down. A time crystal, once it gets going maintains its periodicity.

Mapping it to spatial terms, I think: You could take a bag of beans and lay them out in a nice repeating grid pattern if you want, giving them spatial symmetry. But you have to keep making the grid yourself. The "bean crystal" does not naturally spread on its own. When a real crystal cools, the entire substance aligns into a regular repeating pattern on its own without any external force needed.

Sorry, I misunderstood then. But still, personally I think it's about as relevant as if the post was about dark matter, and someone commented "well, dark chocolate is also quite hard to detect if the fridge light goes off".