Apple doesn't make this easy. There's no Save Video button in the app. Here's the cleanest workaround we've found.
If you don't see Save Video. Tap the download arrow in the bottom-left corner of the preview. The file goes to Files first, then you can move it to Photos.
Audio-only?. For just the song, use the Reels to MP3 tool — the MP3 saves to Files and plays in the Music app.