That wasn't what I got from this video. To me, it sounded more like he's saying:
If people are getting bullied, don't blame the character blame the bully. If it wasn't Apu there would just be something else people would use instead.
Apu may have been created as a stereotype, but when given the chance to focus on his character Apu is shown to be so much more than a caricature.
"We are laughing with him" is to bring home that last point that he isn't simply a funny accent to point and laugh at. The things that happen to him are things people relate to which is why we find them funny, just like other characters in the show.
If you don't agree that he isn't a caricature, then that's something else entirely. But I feel like the video makes a pretty solid argument against that.