Although he's pretty separate from everything that goes down in Matilda, the narrator definitely has opinions about things, like students and parents and teachers and the Trunchbull (for the most part, those opinions are, shall we say, negative). But we don't really learn anything else about him—besides those opinions, that is. He's not in the trenches, fighting the Trunchbull, or pranking papa.
Though the narrator doesn't make a big scene, he does, sometimes, pop in for a visit. When Matilda's narrator brings out the "I" voice in this story, it's so he can share one of his opinions. It's almost as if he can't resist making comments about characters who are so outrageously smart, evil, or awful. He loves to point out the ridiculous, such as when he tells us, "If I were a teacher I would cook up some real scorchers for the children of doting parents" (1.4).
Our narrator is also quite careful with his words. Did you notice that when he talks about Matilda's ma and pa, he often calls them "the mother" or "the father," instead of Mrs. Wormwood and Mr. Wormwood? That's a little strange, right? But we promise—he knows what he's doing. Calling them these terms makes them feel impersonal. This is not a dynamic duo of a doting mother and a faithful father. They're like cardboard cutouts of a mom and dad, and not very good ones at that. The same goes for Miss Honey's awful aunt, whom she calls "the aunt," not my aunt. "The" is just a tiny little word, but it has a big effect when the narrator uses it in this way.
It also gives us a big hint: this narrator is on Matilda's side (even when she's making mischief). He's anti-adult, pro-pupil, and he's not afraid to let us know.