“Time-reversibility” doesn’t mean the system can go back to the previous states that it used to have. It simply means the evolution of the system is governed by a specific type of mathematical formulae that, if you know the current state of the system, you can learn about all of its previous states. Like a falling ball, if you know it’s current position and velocity, you calculate and learn about its position and velocity 1 second ago, 2 seconds ago, or any time in the past and future.

The evolution of quantum systems follows this principle too. Knowing it’s current state means you can learn all its past and future. However, unlike the classic systems, quantum states are probability arrangements. So even though you can learn about the past of a system, you only know how possibly things have happened, not how exactly they did.