This is perhaps a MUCH over-simplified explanation, but here goes.
Essentially all aircraft performance depends upon the 4 basic vectors, Weight, Drag, Lift, and Thrust. These may all be resolved geometrically, which, when there is an excess of thrust, yield a climb angle. Following on from this, all climb performance, when related to the 4 basic vectors indicate that all climb performance is at a particular angle, thus, the amount of excess thrust determines the climb angle.
If we want to take it to the next step, and find the rate of climb, then we must examine the rate of doing work, and,
Power is the Rate of doing work.
and also,
Power = Force (Thrust) X Velocity (TAS).
Therefore, to find the best RATE of climb, we must examine the best excess RATE of doing work, i.e. excess Power.
Regards,
Old Smokey