I'm a bit rusty on this because EE is not my field, but the key thing to bear in mind is that the complex numbers are not real, they are just a mathematical "trick" to help with the calculations.

Think of it like this: the voltage and current are both sine waves, and they can either be in phase, or out of phase, or somewhere in between. The magnitude of the complex number is the amplitude of the wave (volts or amps), and the argument (angle) of the complex number is the phase angle of the wave compared to a reference wave. If two waves are in phase, the angle between them is 0°, and if the waves are out of phase the angle is ±90°. (Think of plotting voltage and current on an oscilloscope and seeing whether they line up.)

If the voltage and current waves are in phase (as with a resistive load) then the power is simply voltage x current. This is intuitive, and it also works out because the cosine of zero is one. So volts times amps times cos 0 = volts times amps.

If the voltage and current waves are out of phase (as with reactive loads like ideal capacitors or inductors), then the power is zero. This also works out because cos 90° = 0. So volts times amps times cos ±90° = 0.

If the angle is somewhere between 0° and ±90° then the power is somewhere in between two, and is given by the formula V x I x cos (phase angle).

The reason for this formula requires a mathematical derivation, and I think you are being asked to take it as read.