The best short answer I can give is that it is because when the pressure increases, the same amount of the gas is squeezed into a smaller area.
Like I said above, the same amount of the gas is squeezed into a smaller area with an increase in pressure.
Remember that pressure is measured by the amount of molecules of the gas hitting the sides of the wall in the container.
Because of this, when the pressure increases, so does the number of molecules hitting the wall of the container, and the best explanation for that is that the walls have come closer in, thus a smaller volume.
This can be proven using Boyle's Law, which is a form of the Combined gas law. The combined gas law states that:
Since you original question asks why when the pressure increases, the volume decreases, so let me demonstrate that for you.
Say we had
So now we have
And your situation says if we increase the pressure, the volume will decrease.
So let's change the starting pressure from 1 atm to 2 atm in the finishing pressure. So now we have
To solve for
So if we increase the pressure of our example to 2 atm, our volume will decrease to 1 Liter.
So there is your explanation of why when you increase the pressure, the volume will decrease.
Hope it helps!