This is for a perfect gas, but we can use it approximately for air and even water! 
PV = nRT
P= pressure
V = Volume
n = number of moles
R = some constant
T = temperature
Keep temperature constant and the right hand side is constant, forcing you to keep the left hand side constant.
Increase the volume, decrease the pressure of the gas in the chamber.
In your syringe, you increase the volume, thus decreasing the water pressure, in 3D!! The air which is present in the water expands, because of that decreased water pressure.... and expands so much that you can see the air bubbles.
Now go back to 8th-grade chemistry.
PV = nRT
P= pressure
V = Volume
n = number of moles
R = some constant
T = temperature
Keep temperature constant and the right hand side is constant, forcing you to keep the left hand side constant.
Increase the volume, decrease the pressure of the gas in the chamber.
In your syringe, you increase the volume, thus decreasing the water pressure, in 3D!! The air which is present in the water expands, because of that decreased water pressure.... and expands so much that you can see the air bubbles.
Now go back to 8th-grade chemistry.
