If we were to use y=mx+b, then the hough transform image would look like many straight lines intercepting at a few points, which makes most intuitive sense. The issue with this form is it gets ill-formed when the line becomes near vertical (m goes to infinity).

The polar parametrization r=x·sinθ+y·cosθ solves this problem, and in the hough space, the axes will be r and θ. A point in image space maps to a sinusoid in hough space, which is why the transformed image looks like that.