Refer to figure.
Learning Objective 032.01.04.01.03: Explain climb performance in relation to the thrust available and thrust required (angle of climb), and power available and power required (rate of climb).

As described by the above learning objective, the rate of climb of an aircraft is directly proportional to the power available minus the power required (i.e. the excess power), divided by the weight. We can discuss how this comes about below.

As we know, and can prove from first principles using the force diagram above, the angle of climb of the aircraft conforms to the below formula

(Ta - Tr) / W = sin (γ) ~= climb gradient
Where Ta = Thrust Available, Tr = Thrust Required, W=Weight, γ (Gamma) = Angle of climb.

Power = Thrust x velocity, so both Ta and Tr can be multiplied by true airspeed (TAS) to give Pa and Pr (Power Available and Power Required).

The rate of climb is the climb gradient multiplied by the velocity, so therefore by multiplying both sides of the equation by velocity, we get the following.

(Pa - Pr) / W ~= Rate of climb

Therefore, the excess power divided by weight is approximately directly proportional to the rate of climb of the aircraft.