Having struggled to see the world as Bishop Berkeley thought it should be seen, as colors on an essentially "flat" background, Stephen allows his mind to snap back into its normal perception of depth: "Ah, see now! Falls back suddenly, frozen in stereoscope. Click does the trick." The stereoscope was a 19th century invention, precursor to the 3D movie glasses of today, that allowed people to resolve a pair of identical two-dimensional images as a single three-dimensional scene.
The stereoscope, the first version of which was invented by Sir Charles Wheatstone in 1838, reproduces the effect that human eyes (and the eyes of other predators oriented in a single plane) produce at every waking moment: the effect of depth. It directs both eyes toward separate images but places those images in the same location, encouraging the brain to meld them into a single image infused with depth and the consequent sense of roundness and solidity.
If it seems a bit odd for Stephen to have to call upon the aid of a clever artificial device to see the world in the way that all creatures with binocular vision normally perceive it, the reader may reflect that Proteus began with him "walking into eternity along Sandymount strand" by closing his eyes, only to decide upon opening them again that the physical world was "There all the time without you." His experiment with seeing reality as flat is of a piece with his metaphysical effort to pierce through the veil of appearances to the underlying truth.