Yes, it was a parody of then-popular entertainment about World War Two, and several of the performers had suffered persecution by the Nazis. And it probably was funnier in its time.
But taken on its own, why does it offend people? For starters, the people are just too stereotypical. Sergeant Schulz is just too stupid. But what certainly offends people most is Colonel Klink. He's supposed to be a vain camp commandant. Unfortunately, he's quite unconvincing as a villain. He's too nice and genial, and this personality isn't a mask disguising a monster. He's about as realistic as a Nazi officer as Elmer Fudd. He's completely nonthreatening and doesn't seem capable of harming a fly. But he's not a secret anti-Nazi either. The show could have been better if Colonel Klink had shown at least a little bit of menace.
But taken on its own, why does it offend people? For starters, the people are just too stereotypical. Sergeant Schulz is just too stupid. But what certainly offends people most is Colonel Klink. He's supposed to be a vain camp commandant. Unfortunately, he's quite unconvincing as a villain. He's too nice and genial, and this personality isn't a mask disguising a monster. He's about as realistic as a Nazi officer as Elmer Fudd. He's completely nonthreatening and doesn't seem capable of harming a fly. But he's not a secret anti-Nazi either. The show could have been better if Colonel Klink had shown at least a little bit of menace.