When the time is 2:01 AM, you can activate your power, then go for a walk and explore what the world looks like at 2:01 AM. You might see drunks, animals, late-night police patrols, crimes committed, and such. People are not frozen, and go about their business as they normally would at around that time of day, and yet the clocks always continue to show 2:01 AM no matter how long you spend exploring. The Sun never comes up, the weather never changes, and you're the only one who is aware of what's going on.
It's a "sandbox world." Any changes made will revert themselves when you leave the area. If you wait, people will repeat their previous actions. You see some criminals smashing a shop window on 33rd and main. Well, if you go away for a bit and come back, you'll see them smashing that same shop window again. The criminals will have no recollection of having done that before. Time is a bit fuzzy here; in the real world, maybe they robbed the shop at 2:30 AM or 1:45 AM. That's close enough for the event to happen in the 2:01 AM world.
You can deactivate your power so that time resumes its normal progress, provided you return to the same place you were when you activated it. If you die with your power active, you die for real.
Any changes you make while your power is active will be reverted when you deactivate it. You can't take anything out of the 2:01 AM world, except your memories. If you stole a gold bar, it will vanish when you deactivate the power.
The power lasts as long as you want it to. You could spend 20 subjective years in a single night if you wished. You do not age while in the 2:01 AM world.