Solution:

A phase front moves along the fiber perpendicular to the direction of travel.

The wave front must be in-phase with itself across the diameter of the field.

Consider Figure, As the phase front moves into a bend the light at the inner radius of the the curve must move more slowly than the light at the outer radius (considering a single wave occupying all of the mode fields).

This means that at the outer edge (of the core) the light must experience a lower RI than it would in a straight fiber.

If the bend becomes tight enough that the apparent RI of the edge of the core will lower to become equal to the RI of the cladding; thus, the wave is no longer confined and will escape from the fiber.