A unitary 1×1 matrix is a complex number z such that , in other words a complex number of modulus 1. This represents a point on the unit circle in the Argand diagram (that's where the circle comes in), and it is of the form .
An element of SO(2) is a 2×2 real orthogonal matrix with determinant 1. Any such matrix must be of the form , and it represents a rotation through an angle .
The isomorphism from U(1) to SO(2) is the map that takes to .