there are 2 ways to get DLC to work on your console
1. the content is originally downloaded on that console. this way you can play it on any gamertag on that console and even play it when you're offline.
2. signed in with the gamertag that downloaded it. with this, you can play your songs on a different console than that it was downloaded on. downsides are that you have to be signed in with the gamertag that downloaded it (although it doesn't have to be the person playing, you can sign it in as player 2 and not let it join) and you have to be online.
now if you did download it on another console, you can get it to work as option 1 by doing a license transfer in your system settings. this will set all your dlc to that console. downside is that you have to redownload EVERYTHING you downloaded to get it to work. (all these downloads will only take 1 second though, the data is already on your HDD and it really just needs to re-verify it. but still a pain to do)
my love is like a candle, if you forget me, i will burn your damn house down.