Symbolism, Imagery, Allegory
In Star Wars, a Jedi apprentice (Padawan) wears a small braid. Auggie spent many Star Wars-obsessed years growing his own Padawan braid, and he and his best childhood friend Christopher each had a braid complete with beads they had chosen together. Alas, the braid has to go after the first day of school because it turns out to be a lightning rod for unwanted attention from Julian.
It takes Auggie a while to be able to articulate the idea, but he realizes it is time to rework his image now that he is in middle school—and part of that process involves not being a one-note Star Wars kid. The Padawan braid represents Auggie's younger childhood interests and fantasies. His friend Christopher has moved away, their friendship has drifted into the past. Auggie's concerns are now much more to do with the real world than with his Jedi training, and whacking off the braid signals a concrete, intentional separation from those years.
In Star Wars, Padawan's cut off their braids when they become official Jedi Knights, so we can also see Auggie's decision to cut off his braid as symbolic of him coming into his own. He is ready to find his place in the world. And middle school is where this next leg of his journey begins.