Why are we able to see air bubbles under water?

Air and water are both transparent to a good enough approximation. However, light travels more slowly in water: the speed of light in air is about 33% faster than in water. As a result, when light passes from one medium to the other, it is partly reflected and partly refracted (bent). For the refracted part, the general rule for determining the bending angle is called Snell's law, which can be expressed like this: $$ \frac{\sin\theta_\text{w}}{\sin\theta_\text{a}}=\frac{v_\text{w}}{v_\text{a}} \approx \frac{1}{1.33} \tag{1} $$ where $v_\text{w}$ and $v_\text{a}$ are the speed of light in water and air, respectively, and where $\theta_\text{w}$ and $\theta_\text{a}$ are the angles of the light ray relative to a line perpendicular to the surface, on the water side and on the air side, respectively.

If the angle on the water side is $\theta_\text{w} \gtrsim 49^\circ$, then equation (1) does not have any solution: there is no air-side angle $\theta_\text{a}$ that satisfies the equation. In this case, as niels nielsen indicated, light propagating inside the water will be completely reflected at the water-air interface. So the rim of the bubble acts like a mirror: if you do a reverse ray-trace from your eye back to near the rim of an air bubble in the water, the angle between the ray and the line perpendicular to the surface of the bubble will be greater than $49^\circ$ (this defines what "near the rim" means), so that part of the bubble acts like a mirror for light coming from those angles, as illustrated here: enter image description here

You can see light reflected off of the surface of a submerged bubble because the index of refraction of the air inside the bubble is different from that of the water that surrounds the bubble.

That difference, if great enough, will turn a bubble surface into a mirror for light rays that approach it from certain directions, thereby making it easy to see.

This condition is easily met for the combination of air and water.

A google search on "refraction" and "total internal reflection" will furnish more examples of this, and explain the math behind it.