Given a continuous function g, restrict g to the boundary of the disk. This is f, and it's continuous. In other words, f defines a loop running around S1. For each radius r, evaluate g on the circle of radius r. This is another loop from S1 to S1. A homotopy runs r down to 0, making f and the constant function g(0) homotopic. These have the same degree, hence f has degree 0.
Conversely, degree 0 means a homotopy h shrinks f to a point. Apply this homotopy as the radius shrinks from 1 to ½. Now g is constant on the circle of radius ½, and we can set g equal to this constant for r < ½. (Carrying the homotopy from 1 to ½ makes the continuity proof a little simpler.)