Here's the way I look at it:
Vy is the speed of maximum excess power = power available - power required. If power available were not a function of speed, that would be the same as minimum power required speed (minimum sink rate for a glider). But since power available increases with speed, Vy occurs at a higher speed than minimum power required. As the power available reduces with altitude, Vy reduces to much closer to the minimum power required speed at the absolute ceiling of the aircraft.
Vx is the speed of maximum excess thrust = thrust available - thrust required (drag). If thrust available were not a function of speed, that would be the same as minimum drag speed (best glide for a glider). But since thrust available decreases with speed, Vx occurs at a lower speed than minimum drag speed. As the thrust available reduces with altitude, Vx increases to much closer to the minimum drag speed at the absolute ceiling of the aircraft.
You ask about the difference between props and jets. Props are closer to constant power systems -- thrust reduces significantly with speed, and so the effect increasing Vx is more significant than the decrease in Vy. Jets are closer to constant thrust systems -- power increases significantly with speed, and so the effect decreasing Vy is more significant than the increase in Vx. Both of those are simplifications: in fact thrust decreases and power increases with speed for both props and jets.
There's one more complication which is that the power required is a function of both IAS/CAS because it depends on drag, and TAS because power is drag times (true) speed. As altitude increases and TAS increases for a given IAS, the minimum power required will reduce. This effect will tend to reduce Vy with altitude, even for an engine that doesn't care so much about the thinner air.