I was brought up listening to most of those on my father’s records, and didn’t see any of them for years after.
I’ll second West Side Story and Rocky Horror, and add HMS Pinafore by Gilbert and Sullivan (even though most of their stuff was just as good).
Oh, does Elvis count? 
I’ll second this. I’ve never seen The Who’s Tommy but I love the soundtrack. I think I have a pretty good grasp of what happens throughout the musical based upon the songs as well.
Pacific Overtures soundtrack tells the story to me.
But I already knew the backstory, so that may be cheating.
Many Broadway recordings are great, usually being recorded after the cast has performed the plat several times.
The Blues Brothers.
Wicked, for sure. I loved the show long before I saw it.
Sunset Boulevard, in its own way. I have the CD, but have never seen a production (or the movie, for that matter) and the story’s perfectly coherent.
A Chorus Line: I hate the show itself. I’ve only seen the movie, which I’m sure has a lot to do with it, but I HATED it. Love the music, though.
Henry Mancini’s soundtracks all have the characteristic that they make great listening whether or not you’ve seen the movie. Some of the more notable ones are:
Breakfast at Tiffany’s
Hatari!
Charade
The Pink Panther
Experiment in Terror
Even the TV series “music from” albums are great that way:
Peter Gunn
Mr. Lucky.
Chess and The Blues Brothers are two that spring readily to mind…
Definitely.
A Chorus Line
Based on this, I have just ordered it. It sounds like something I would love!!
For me, Rent, Phantom, Les Miz (I’ve seen it in Chicago and NYC but loved the recording for years before), West Side Story all hold up on their own.