MAP is short for 'Manifold Absolute Pressure', not "manifold air pressure". It is often referred to as 'Manifold Pressure' (MP).
The greater the pressure in the inlet manifold, the greater the amount of fuel/air mixture that will enter the cylinders. More fuel & air in the cylinders equals more power being produced. 'More fuel & air' in this case means measured by weight, not volume or pressure.
For all practical purposes though, the density of the mixture is constant, so the pressure in the manifold will be representative of how much mixture will go into the cylinders.
Closing the throttle causes the MAP to reduce because the throttle restricts how much mixture can flow into the manifold to replace the mixture that the cylinders have just sucked out. Hence the term 'throttle' - it literally throttles the engine.
Less mixture available in the manifold to enter the cylinders means a lower power output from the engine.
Opening the throttle removes some or all of the restriction, allowing more fuel/air mixture to enter the manifold to replace what the cylinders have sucked out, ready to enter the next cylinder that has an open inlet valve.