The more power wasted on the atmosphere, the slower you go up.
Every second, your engine releases some energy from the fuel it burns. Because of the Conservatrion Of Energy, it has to go somewhere. There are three obvious places:
1) Your aircraft can go faster, i.e. get Kinetic energy. But you are climbing at a constant speed, so no energy goes here.
2) Your aircraft can go higher, i.e. acquire Potential Energy. This is what you want.
3) The atmosphere can be moved about, i.e. acquire kinetic energy. You are right, this isn't really "wasted". I should have said it is an inevitable consequence of moving the aircraft through the air (parasite drag) and generating lift (induced drag). Parasite Drag involves a random swirling of air (Kinetic Energy but no momentum), and Induced Drag involves a downwards directed flow of air (Kinetic Energy and momentum).
The energy goes partly to 2, and partly to 3. The more that ends up in 2, the better for your climb.
Incidentally, if you are happy dealing with vector components of forces, there is a more direct explanation for "Excess Thrust".
Consider first a glide approach. Lets say Thrust is actually zero. Drag is still there, and because you are not accelerating (just moving steadily downwards) Lift equals Aircraft Weight (very closely) and Drag is Weight divided by the Lift/drag ratio.
Say LDR=10 so Drag is 10% of aircraft weight. If you descend at 6 degrees, (1 in 10) then 10% of Gravity (ie Weight) is pulling you along and making up the "Thrust Deficit". 10% of say 65kts is about 650 fpm so this makes sense.
If you extend flaps or the prop is windmilling the LDR might be 8, and you need to go down at 1 in 8 to overcome the Thrust deficit and maintain speed. This is about 800 fpm.
In the climb, you need Excess Thrust to overcome a component of Gravity which still depends on your angle of climb, but now works against you.
And I agree with Piper Classique, "See How it Flies" is a wonderful resource!