1. The Structure of the Fiber: An optical fiber consists of a "Core" with a higher refractive index ($n_1$) surrounded by a "Cladding" with a lower refractive index ($n_2$).
2. Total Internal Reflection (TIR): When light traveling in the core hits the boundary with the cladding at a shallow angle (specifically, an angle of incidence greater than the critical angle), it does not pass through into the cladding. Instead, 100% of the light is reflected back into the core.
$$\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)$$
3. Propagation Mechanism: This process of Total Internal Reflection occurs repeatedly as the light travels down the fiber, zig-zagging back and forth but always staying contained within the core. This allows the fiber to "pipe" light even around bends and curves without losing the signal to the outside environment.