Let a homotopy move the base point c around in S, and return it to C, thus forming a loop. Build a new homotopy, which follows the old, but leaves a tail back to c. At t = 1 you have the desired image of the sphere, along with a loop from c back to c. Let another homotopy shrink this loop down to c. Thus the desired image remains homotopic, with c fixed throughout time.
Next, let the base point start at c, but wind up at b. Leave a tail behind at all times, connecting the base point to c. The image based at b is now homotopic to an image based at c.
Put this all together and every image is homotopic to an image based at c, and every pair of homotopic images based at c can be equated by a homotopy that leaves c fixed throughout time. Therefore πn(S) gives the homotopy classes of the n sphere in S.
The analysis is a little more complicated when S is not simply connected. I'll save that for another day.