If light is radiated in all directions why does an image of a remote star converge on an eye-piece?

Light from a distant object does not "converge" on an eye piece. What happens is that some amount of the light emitted is intercepted by the eyepiece, based on the cross-section area of the eyepiece and the distance from the light source. Hence the further away from the source you are the less of the light emitted by the source is collected by the eye piece, and hence the dimmer the source appears.

For "reasonable" distances from the light source, the fraction of light emitted by the source that impinges on the eyepiece is



where = area of eye piece and = distance from source to eye piece. Note that if doubles, then the amount of light captured by the eyepiece falls to 1/4 of what it was.