You checkin online for your entire trip the day before you board the train in Stuttgart. In Frankfurt there is a separate airrail desk to check your bag on the way from the station to the terminal. In T1 there is a separate security lane for airrail passengers to accommodate the short connections. Ask at the airrail checkin desk when you give them your luggage.

on the way back the airrail luggage shows up at a belt opposite the airrail desk so you don’t have to carry your luggage all the way from terminal to the station.