Open the .dmg and drag Butter Motion into Applications.
Open Butter Motion from Applications.
The app opens on a Sign in screen. Click Sign in via browser, approve this computer in the browser tab that opens, and come back to the app.
Click Install. The app downloads and sets up the plugin inside Premiere, moving through stages from Downloading to Starting.
When the app shows Everything's ready, restart Adobe Premiere Pro. The panel only appears after a full restart of Premiere.
Check that it worked
After installing, open the panel in Premiere:
Open the panel in Premiere via Window → Extensions → Butter Motion. Since you signed in inside the app, the panel should already be signed in. If it isn't, use its own Sign in via browser button, see Sign in & connect a device.
If the panel doesn't show up, quit Premiere completely and reopen it. If it's still missing, see Panel missing in Premiere.