Here is an answer which would make sense to an elementary school student—if they understood what you were asking. It's Grassmann's original argument for considering anticommutativity. I don't have a reference handy, but I'm pretty sure it shows up in the intro material to one of the Ausdehnungslehre, or perhaps a summary essay.
Grassmann's goal was to find a way to "arithmetize" geometry. So, let's do that very naively. Suppose you have a line segment AB and another collinear line segment BC:
Then, through visual inspection, AB + BC = AC. However, now suppose C lies in the middle instead of B:
Writing down the obvious equation from this arrangement we get AC + CB = AB
If we solve the resulting system of two equations, we realize that BC = -CB
An additional thought for Qiaochu, have you looked at Klein and Rota's book Introduction to Geometric Probability? There are some interesting analogies there between combinatorial structures and geometry that might give you some thoughts. In particular, they link inclusion-exclusion and the Euler characteristic as the unique 0-dimensional invariant valuations in the combinatorial and geometric settings respectively.