This is the crux of the observation. For needles of length less than W, the probability that it crosses a floorboard is equal to the average number of floorboards it crosses. (Exercise for the reader ;))

The point is that the "right" quantity to be considering for the problem is the average number crossings, since that naturally extends to curved noodles, lines of any length, and even circles. The number of crossings is also known as the Euler characteristic of the intersection, and there's a rather deep and beautiful theory of geometric probability that takes this as the jumping off point.