You were able to play his content previously because the Xbox 360 uses a license system that is bound to the console.
Your brother will need to do a license transfer to the new console:
- Sign into the new Xbox with your brother's account
- Go to settings, and then select Account
- Go to Your Billing Options and select Licence Transfer
You will need to redownload the games again in some cases (this will download the new license details per game).
If your brother still uses the old console he won't see any difference as long as he plays his games whilst connected to Xbox Live. If he disconnects he will not be able to use some of the content (if he is mid-game, he will get a 10 minute countdown before the game automatically closes). Note that this refers to download content such as DLC and games on demand titles. Playing from disk is fine.
There is a cool-down period for license transfering. Currently, you need to wait 4 months between each transfer.
You can find more information from Xbox's official site:
Xbox 360 DRM Details