Well, from the perspective of everyday life—way, way, way above the quantum level and a lot more complicated—the magical measuring tool is that we send some photons toward the half-silvered mirror, one at a time, and count up how many photons arrive at Detector 1 versus Detector 2 over a few thousand trials. The ratio of these values is the ratio of the squared moduli of the amplitudes.

The ratio of "photon at detector 1" and "photon at detector 2" (averaged over enough trials) is 1.

If none of that made sense, don't worry. It will be cleared up in future posts.

Edit: This was actually written as a response to one of these comments.