Most aerodynamics books include the appropriate Drag/Thrust and Power available/required curves, but for many people these are far from intuitively obvious.
VX is the speed at which achievable angle of climb is greatest. We can get a mental picture of why climb angle is related to thrust and drag, if we consider two extreme cases
In straight and level flight the lift acts straight upwards. So all of the weight is carried by the lift. This means that the thrust required is equal to the drag.
In a vertical climb any lift would act horizontally, so all of the weight must be carried by the thrust. But we still have drag, so the thrust required is equal to the drag plus the weight.
As climb angle increases between zero (straight and level) and 90 degrees (vertical), the proportion of the weight that must be carreid by the thrust gradually increases from 0% to 100%.
The proportion of the weight that is carried by the thrust is equal to the weight times the sine of the climb angle. So at any climb angle, (including zero) the required thrust is equal to the drag plus the weight times the sine of the climb angle.
At zero degrees climb angle, the sine is zero so thrust required = drag.
The sine of 90 degrees is 1, so at 90 degrees climb angle the thrust required = drag plus the weight.
This means that the climb angle that we can acheived is proportional to the amount by which thrust exceeds drag. So our best climb angle is achieved at the speed where thrust exceeds drag by the greatest margin. To find this speed we need simply plot thrust and drag against speed, then find the speed at which the excess thrust is greatest.
VY is the speed at which rate of climb is greatest. To understand why this is related to power we need to start with some basic definitions.
Work is done when a force moves its point of application in the direction of the force. The amount of work done is equal to the force applied times the distance moved.
When an aircraft is in straight and level flight, the thrust must equal the drag. So the work done is equal to the thrust (or drag) times the distance flown through the air.
Power is the rate of doing work, which means that it is equal to the work done divided by the time taken to do that work.
So for our aircraft in straight and level flight, the power being used is equal to the thrust x distance flown/time taken. But distance/time = TAS, so power being used = thrust times TAS.
The power that is available for straight and level flight at any given TAS is equal to the thrust times TAS. And the power that is required is equal to the drag times the TAS. Provided thrust is equal to drag in straight and level flight at any given TAS, then we have sufficient power available to fly at that TAS.
If we now want to add a vertical speed component (ROC) we must provide some additional power to push the aircraft upwards. The force required to push the aircraft vertically upwards = weight and the vertical speed is the ROC. So the additional power required = weight times ROC.
This means that our best achievable ROC is proportional to the amount by which our power available exceeds our power required for straight and level flight. The greater the excess power available, the greater will be our best ROC. VY is the speed at which this excess power is greatest, so flight at VY gives best achievable ROC.