Browsing another disk in Time Machine just lets me browse my actual hard drive I'm clicking "Browse other backup disks..." in the Time Machine menu, then selecting a Time Machine disk. This puts me into the Time Machine interface, but I just see my home folder. Not my home folder from my old computer, but my current home folder. How can I browse my old backups to retrieve files?
To retrieve files from a Time Machine disk without using the Time Machine interface, open a new Finder window and open the volume from the sidebar. You should see a folder called Backups.db. Open this and the folder it contains for your computer.

Note: If you're using a Network Disk, such as a Time Capsule or Airport Extreme with a connected disk, you will not see Backups.db, you will instead see a .sparsebundle file. Open this and you should be able to continue these instructions.

You will now see a list of folders, named by their date/time. Open the folder you wish to retrieve the file from and navigate to the appropriate location.