Question

how we can see the air bubbles

Answer 1

The bubbles form as air is trapped in the water.  Surface tension separates the air and water; it acts all over the surface, exerting a force toward the center of the bubble: this force is resisted by an opposing pressure of air in the bubble.

Air is soluble in water, and the solubility of air in water follows hennery law "the amount of air dissolved in a fluid is proportional to the pressure in the system". The amount of air that can be dissolved in water increases with pressure and decreases with temperature of the water. Therefore, when we pump air through water, the air is at a higher pressure than water and so the un-dissolved air come out as air bubbles.