First recall below a beautiful geometric proof of the law of reflection i.e. incoming angle=outgoing angle using Fermat's principle by flipping the outgoing path in the mirror plane:
$\uparrow$ Fig. 1 The law of reflection. The shortest path between $A$ and the mirror point $B'$ is a straight line. (Image from the quantapublication.wordpress.com website.)
Obviously, the shortest path is a straight line. Or equivalently, the straight line is a stationary point of path lengths in the set of all paths.
On one hand, the path lengths of neighboring paths to the straight line can only vary in the second order of the deformation. Therefore the corresponding phasors all point in approximately the same direction, and their sum adds up, cf. group-I in Fig. (b).
On the other hand, away from a stationary point, the path length of neighboring paths typically vary linearly in the deformation. The corresponding phasors points in very different directions, and their sum typically cancels, cf. group-II in Fig. (b).
It should be stressed that above observation is at the core of why the path integral is dominated by classical paths, cf. e.g. this Phys.SE post and links therein.