The basic idea I like is, take x^2 + y^2 = z^2, but divide by z^2 to get (x/z)^2 + (y/z)^2 = 1. Now you're looking for the rational points on the unit circle. An important observation is that any two rational points define a line with rational slope. You know of one such rational point on the unit circle, say, (-1, 0). Now you can invert this process and pick a line with rational slope that goes through (-1,0). All lines with rational slope going through (-1,0) will hit another rational point on the circle, so you've found all rational points on the circle by this method, which include all primitive Pythagorean triples.
When you work out this math, you recover the classical parametrisation, where the slope m/n of the line corresponds to the parameters, and this line-intersection method is a good inspiration for how to do the same with some higher-order curves such as elliptic curves, where it leads to the group law on them.