Xbox 360 - Family sharing doesn't work

I did install the games BEFORE i did the license transfer. Do i need to re-download the games AFTER i transffered the licenses?

I am currently playing the games that i cannot play on my new Xbox 360 on my old Xbox 360.

I believe that the profile that has the game must be signed in on an Xbox 360 in order to access the game. It's not like the Xbox One where you can set the console to "home" to share the games with other profiles.

On the 360 it does work a little differently, but you can share games without needing to be signed in. When you purchase a game you're given 2 licenses, a console license and an account license. The account license allows you to sign in and use your content
on any console. The console license allows you to use your content without having to be signed into the console. Anyone who is using the same console you originally bought the content on can access your games. The license transfer moves the console license
if you get a new console so you can continue using content offline or share it.

Me any my brother have had an Xbox since 2009, we got lots of games through Games with Gold that we kept and both of us could enjoy them.

Recently we got a new Xbox 360, the storage was wiped and we downloaded our profiles and some of our games on it.

Turns out we can't play each other's game anymore. We tried transferring the licenses to the new Xbox but it still wouldn't work.

What is the reason behind this error and is there any way to resolve it?

Family sharing is a "Xbox one" feature. However you can still share downloaded content on an x360 if either the content is downloaded onto the "first" console the license was downloaded to, or if the gamertag who purchased the content is logged in.

When you move to a new Xbox 360, you will have to remove the license history from the old console and re-download titles to the new machine for the license transfer to work.

Read this: https://support.xbox.com/en-US/xbox-360/store/download-content#aac7a94c7c404d7a85195c6492ad0a2e

Xbox 360 game sharing works differently to Xbox One game sharing. You'd need to transfer the licences to your new console as Smwutches said.

https://support.xbox.com/en-US/xbox-360/store/download-content#transfer

Did you redownload the games after you performed the license transfer? That would redownload the licenses and you should be able to continue sharing the games.