You give the transit hotel system your flight information. They put you in the most appropriate hotel, usually the one in your arrival terminal. Be sure to note which you're booked into. (You could probably request a specific one, but I've never felt the need to.)

Keep in mind that you won't have access to your checked bags at the transit hotel even if they're short-checked to SIN. You'd have to clear immigration to get them. Unless you already have your boarding pass for the next leg, you then won't be able to re-enter the airport to reach the hotel - and you can't just check in for that leg, because it won't be on the same day. So, carry on whatever you'll need for the night and the following morning.