If you have SMBX2 installed already (and it's the latest version), you can skip this step. This guide just goes over a standard episode installation.
Battle Arena runs on an engine called SMBX2, which you'll need to install first. You can download the installer at codehaus.moe. Make sure that you install into a regular folder, rather than something like OneDrive.
First, download Battle Arena itself. The download is on the front page. Then, find where you installed SMBX2, and go into its "worlds" folder (or "data/worlds"). Move the zip file for Battle Arena into this folder, and then extract it.
Now, after starting SMBX2, Battle Arena should be recognised in the launcher! You can already start it and play local multiplayer. However, online multiplayer takes a bit more setup, which you can follow the guide below for.