Why the formation of rainbows require a total internal reflection on the part of light. I mean can't the rainbows be formed without it , simply through refraction by the water droplet at a suitable angle?
Rainbow formation does not involve Total Internal Reflection in any way. In fact, TIR is impossible in spherical raindrops. Light cannot enter them at an angle (relative to the surface normal) that is greater than the critical angle, and the angle a ray makes when it encounters the surface from the inside is the same angle it had when it entered.
A rainbow is an example of a caustic, which is when reflection and/or refraction causes a wave front to fold back upon itself. The light becomes extremely bright at the point where it reverses. Other examples include star-like patterns when light shines through a goblet of water, and the bright lines on the bottom of a swimming pool in sunlight.
A reflection is required, tho. The folding doesn't happen without it.