Light is incident on the boundary between core and cladding at an angle greater than the critical angle.
Total Internal Reflection (TIR) is a phenomenon that is observed at the boundaries of different wave media. Whenever a wave crosses a boundary between media some of the wave is refracted across the boundary and the rest is reflected. As the angle of incidence is increased more light is reflected and less is refracted. The angles of refraction and reflection also increase.
At a specific angle of incidence (the critical angle) the refracted ray is 90º to the normal line. This means that the ray is parallel to and along the boundary itself. If the angle of incidence is increased beyond the critical angle all of the wave is reflected and none is refracted. This is TIR.
There are actually different types of fibre: step index, graded index and mono mode.
These fibres have a sharp change in refractive index between the core and cladding. The incident light is directed into the fibre so that it reflects off of the boundary at an angle greater than the critical angle just like a mirror. Each subsequent interaction with the boundary along the fibre occurs at an angle greater than the critical.
These fibres have a gradual change in refractive index from the centre of the fibre to the edge. There is no reflection per se. Rather the light gradually changes its direction as it moves toward the edge of the fibre. Light rays follow a sinusoidal path along the fibre. Signal quality is better than for Step Index fibres.
These fibres have a tiny core that effectively concentrates the light into a single path. There is no significant sequence of reflections at the edge. These fibres have the signal quality.