It was one of the first Jim Henson movies I've ever seen, and the line "You have no power over me" still gives me goosebumps every time I rewatch it. I can't quite put my finger on it, but most of Henson's movies have an unique charm to them and these puppets have so much personality in them that you can almost forget they're controlled by puppeteers. It's sad that most movies nowadays rely heavily on CGI and digital extra effects when practical effects can feel so much more "real".
I'm glad I'm not the only adult that can forget those are puppets sometimes. When I'm sinking in and immersed, they're just characters as real as the humans (though I apparently have something called hyperphantasia, which makes that pretty easy for me).
And, I agree, it really comes down to the Henson design and the skill of the puppeteers.