This advice may be too late. Usually for Amtrak you want to buy luggage tags ahead of time. Basically the announcement means that if they see a bag that doesnt show ownership by anybody or a phone number to call, then they're worried that it might be a auitcase bomb. I don't know if Amtrak has free tags at their ticket counter or not. You may be able to find luggage tags at those newsstand shops that sell travel pillows and stuff. But those tend to be more expensive. I use these.

You can also ask a Red Cap for help, altho they need to be tipped. I usually give them $5 coz having an exoert's help is worth the cost to me. In DC Union Station, the Red Cap stand is found across from Gate D. Gate D is also where the Club Acela lounge can be found if you have an Acela or sleeper car ticket.

Eta: I realize now that I answered a question that you didn't ask, and I'm sorry for that. When you find the Gate listed for your train number, they open that gate maybe 30m to an hour prior to departure. Often folks will line up outside it prior to them opening the gate. I know it's overwhelming the first time. A Red Cap is happy to help, or any train official standing close to the gate or at a ticket counter