Why can't my brother play the games I bought on my account?

Summary:

You need to transfer the content license to his Xbox.

Details:

After reading over Xbox Support's page on Digital Rights Management, and comparing to my own experiences as a 360 owner, I believe that you can play a downloaded game on a 360 console if you have the appropriate content license. This is the case if either of two conditions is true:

1. It's the console with the local license for the game. By default, this is the console to which the game was first downloaded.
2. You're using the account with the system license for the game. This is the account that purchased the game.

So, your brother can play the game on your Xbox using his account, because it's the Xbox to which the game was initially downloaded. It has the local content license. Also, you can play the game on your brother's Xbox using your account, because it's the account that initially bought the game. It has the system content license.

But, in order for your brother to play the game on his Xbox with his account, you would need to transfer the license.

From the DRM page:

Why can't my friend play the games I downloaded to his/her console?

The license for content you purchase is for a single account or a single console. After you purchase and download an Xbox Live game, anyone in your household may use it. You can install and play the game on other consoles. However, only the account that purchased the game can play it. A friend can't play the game using his or her own account. To do so, your friend should purchase the game using his or her own Xbox Live account.

and

Can I transfer content from one Xbox Live account to another?

No. Content licenses are stored in the Xbox 360 console rather than in the Xbox Live account.

To let someone else use the content license, you must download your Xbox Live profile to their Xbox 360 console. Then, transfer your content license to that Xbox 360 console.

To download your Xbox Live profile to a different Xbox 360 console, see Download your Xbox Live profile to a different Xbox 360 console or redownload it.

To transfer a content license from one Xbox 360 console to another, see above section, Can I transfer a content license from one Xbox 360 console to another?

After you transfer your content, other profiles can use the content on that console.

So you can transfer the rights to your brother's Xbox. This will allow you to continue playing the games on either Xbox, and allow your brother to play them on only his Xbox, but not yours. Keep in mind you can only transfer licenses every four months.

Actually. If you have two xbox 360 controllers, the he needs to sign you in on one and then he can sign in on the other and play single or multiplayer if he has gold then he has the option of local or online multiplayer. But yes it will work I do it all the time for my kids.

From what I understood, you downloaded all your games on the new 360 using your account, so these games can only be played by that account used to download the games. Even if they are downloaded on the harddrive, only that account can play them, so your brother can't play them using his own account. This is due to DRM (Digital Rights Management):

What is "digital rights management"?

Content providers such as the Xbox Store use digital rights management technology to control the usage and distribution of games and other content. This content is protected by media usage rights that allow specific uses of that content.

You receive a licence to use content when you purchase that content. The licence has two parts:

The system licence. This licence applies to the gamertag that purchases the content. The local licence. This licence applies to the console to which you first downloaded the content.

Solution: simply have your brother play using your account, so he'll have access to all your games.

For more information: Xbox Support - Digital Rights Management