[joppy]

The 4x4 represents both a 3x3 matrix A and a translation t, so that the “affine transformation” applied to a vector is v -> Av + t. One could pass around pairs (A, t) instead, and figure out how to compose those pairs, but it turns out there’s a nice way of embedding them in a 4x4 matrix such that composition is just 4x4 matrix multiplication.

You may opt into some other mathematical niceties too, which are handy for applying perspective transformations, but the main takeaway is that you get efficient compositions of affine transformations.