If you want to understand this at an intuitive, geometric level, you should think of both groups as the group of symmetries of an equilateral triangle.
Any symmetry of the triangle can be described by the way that it permutes the vertices, and any permutation of the vertices can be achieved by a symmetry. So the group of symmetries is isomorphic to S3.
Slightly less obviously, the group of symmetries is generated by two elements, a rotation through radians about the centre, and a reflection in a line joining a vertex to the midpoint of the opposite side. These two operations satisfy the defining relations for the generators of D3.