I created my own Unmanaged Package on a test sandbox. I would like to install the package on another sandbox now, but I am not able to see where to do this from. I have produced the Installation URL. I go to Setup -> Installed Packages and I can see that the only way to install packages is from the AppExchange. Is there a button or something I am missing
Just navigate to that installation URL and it should install fine. I don't know where you saw that you can only install packages that are listed on the AppExchange. In the original sandbox, when you made the package, it created an installation URL on each version. Just navigate there and sign into the other sandbox.
Oh, substitute "test" for "login" in the installation URL. That way you can put it into a sandbox and not production.