After you’ve downloaded it you need to get your legally acquired game ROMs on to your console. The easiest way to do that is to use the FTP app that is on the same link as the Retrospection app is.
first you have to go to settings and do the update assets and update cores options.
Download that FTP app, open it and click set local path, then click start down the bottom.
On your pc, download an FTP program - I like FileZilla - and connect to the local IP address that the FTP app on your Xbox shows on the screen.
FileZilla will now be connected to your Xbox! Go into the downloads folder on your Xbox via FileZilla, make a new folder for GameCube etc. copy paste your gamecube roms into there. Supposedly you can’t have more than 16gb per folder, so just try a few games at a time.
for some emulator Cores you’ll need a bios dump from the actual console that you own. Using filezilla copy them to the System folder in retrospection on your Xbox.
now in retrospection on your Xbox you should be able to go Load Core, choose Nintendo - Dolphin / Wii, then Load Content, go to download folder/GameCube and choose your rom, and away you go