Step 1: Idea of total internal reflection.
Consider a ray of light going out of a dense medium (say water) into air. Because air is rarer, the ray moves away from the normal. As we tilt the ray to larger and larger incidence, the emerging ray leans more towards the water surface. The special incidence at which the emerging ray just skims the surface (refraction angle \(90^\circ\)) is the critical angle \(\theta_c\). Beyond this incidence the surface behaves like a perfect mirror and reflects the entire beam back inside. This mirror-like return of the whole beam is total internal reflection.
Step 2: Getting the critical angle.
Snell's law from denser (index \(\mu\)) to rarer (air, index \(1\)) at \(\theta_r = 90^\circ\) gives \(\mu\sin\theta_c = \sin 90^\circ = 1\), hence \(\sin\theta_c = 1/\mu\). So a denser material (larger \(\mu\)) has a smaller critical angle, which makes TIR easier.
Step 3: Two must-have conditions.
(a) travel from denser to rarer medium, and (b) incidence angle larger than \(\theta_c\). If either fails, ordinary refraction takes place and light escapes.
Step 4: How an optical fibre uses this.
A fibre is made of a high-index glass core surrounded by a low-index cladding. Light fed into the core hits the core-cladding wall at an angle above the critical angle, so it is totally internally reflected. Bouncing wall to wall thousands of times per metre, the signal is trapped inside the core and is piped from one end to the other with negligible leakage, even around bends. This is why fibres are used for high-speed communication and for endoscopes in medicine.
\[\boxed{\sin\theta_c = \tfrac{1}{\mu};\ \text{fibre = high-index core + low-index cladding guiding light by TIR}}\]