How to use the installation disk

I haven't bought a physical disk for Xbox since 2013.

Stick it in the machine I guess...

And I’m confused on what to do

OP, you are not crazy, I had the same exact problem on the Xbox. For some reason, the installation procedure is messed up on the Xbox in particular, it will default to pulling the Mass Effect 2 and 3 installs from the internet if your Xbox is online, and will never let you use the second disc. Here is the procedure (which the game NEVER tells you about by the way) to get the installation disc to work on the Xbox One. I've heard the process is slightly different on the Series X, but I only have a One X to try on. It is the complete reverse of the Playstation's installation procedure where you install the installation disc first.

1. Disconnect the Xbox from your network. This way the Xbox won’t default to pulling the installs (86GB if I recall correctly) from the internet.

2. Install the Play Disc, which installs Mass Effect 1 as well as the Legendary Edition’s Launcher.

3. Once the Play Disc finishes, launch the game and select install for Mass Effect 2 and Mass Effect 3. It might look like nothing happens, but that’s only because unpatched (I assume), there is massive lag on the launcher. About 15-20 seconds after you click on install for either game, you will see the progress bar appear for it. I say go ahead and do both if you have the space, just for efficiency.