Just as a line segment in the plane can be extended indefinitely in either direction by laying down overlapping segments, so too can a geodesic be extended on a curved surface, and this extension is unique.
Tristan Needham Visual Differential Geometry
The intuition given later was that the surface in a way decides how the segment bends but could a more 'careful' intuition be given? In sense, I want something that is more specific as to what is happening.