The town museum occupies the preserved east wing of the former monastery, which makes it the natural follow-up to the Stiftsruine. Its rooms cover the abbey, Bad Hersfeld's civic history, rural life, coins, seals, models, and older town crafts.
It is a small museum rather than a blockbuster, but it gives the ruin some useful context. Look especially for the material connected with the monastery precinct and the old town, since those are the stories you keep seeing in stone outside.