I'm not an FI, but by chance I was reading up on exactly this today.
Excess power matters for Vy because:
At max throttle, for any given speed, your engine produces constant power (ignoring tedious air density effects). This means some thing or things are steadily gaining energy over time. In this case, flying at a constant speed, your aircraft is NOT gaining kinetic energy (ignoring the tedious air density effects again).
However it is gaining height, which is potential energy. The more power that ends up in this category, the faster you go up. This is "excess power", and it is nice to have some.
As your wing generates lift and drag, you also disturb the air you pass through, handing out kinetic energy to the atmosphere at a steady rate. The more power wasted on the atmosphere, the slower you go up.
For aeroplanes, the power available from the engine depends on airspeed. Power lost to the atmosphere also depends on airspeed. There is a speed, Vy, at which you have the maximum difference between the two, leaving the most "excess power" left over to climb at the fastest rate.
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The speed for best angle of climb is more complicated.
It also depend on the excess power, ie engine power minus power lost moving the atmosphere around.
You are trying to maximise Rate of Climb divided by TAS.
As we have seen, Rate of Climb is proportional to excess power, so if you plot Excess Power (Engine Power minus power lost to the atmosphere) versus TAS, the point on the graph with the steepest gradient gives you the best ratio of Climb Speed to TAS.
The reason eg Trevor Thom talks about excess thrust is that for a physicist or an engineer, a Power is a Force times a Speed.
Engine Power is Thrust times TAS.
Power Required is Drag times TAS.
Excess Power is (Thust-Drag) times TAS.
My "max gradient" is Excess Power divided by TAS, ie
Thrust - Drag or "Excess Thrust" as Trevor Thom calls it.
Hope that helps.
PS I see that puntosaurus made some of the same points while I was typing, but I've typed too much already, I think...