The term arises from a common visualization of the complex number x + iy as the point (x, y) in the plane endowed with Cartesian coordinates. When the y axis is oriented vertically, the "upper half-plane" corresponds to the region above the x axis and thus complex numbers for which y > 0.
It also plays an important role in hyperbolic geometry, where the Poincaré half-plane model provides a way of examining hyperbolic motions. The Poincaré metric provides a hyperbolic metric on the space.
Proposition: Let A and B be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes A to B.