I like to think about it from the perspective of continuous time, which is approximated by having #bins = #pixels. The approach relies on having fine-grained bins, since they make error arising from the relationship between binning and the sampling frequency much less significant.

Imagine a point moving along the curve, depositing a constant amount of density/ink onto the canvas per unit of time. When the point is moving quickly it deposits less density, and when the point is moving slowly then it deposits more, since the density deposited per time unit is constant and the point traverses less space when it moves slowly.

You can think of each vertical strip of bins as representing a unit of time. The discrete approximation to arc-length normalization means, for each time series, making sure that it contributes a single unit of density per vertical strip: if a curve goes through one bin in that strip, then that bin has its density increased by 1. If it goes through 3 bins, then the density in each is increased by 1/3.