A day room in jail is just a room that is setup to where the inmates in jail can sit in chairs or at tables and play games such as Chess, Checkers or watch TV.
It's basically like a waiting room and in one way the day room is a waiting room where the inmates are waiting for their time to be up in jail.
The day room usually has a TV or a few TV's in it and the inmates can usually drink coffee, water etc and sit at tables or just talk to each other to pass the time.
It's a place where the inmates can come out of their jail cells in the morning and get some time out of the jail cell.
The inmate would only stay in the jail cell all day if they were a security threat or a dangerous inmate.
But if not then they can come out of the small jail cell in the morning and usually stay out in the day room until around 6:00 PM to 8:00 PM depending on what time the inmates need to be in their cells for bed.