Why does Vx have a relationship with thrust but Vy has a relationship with power?
The angle a plane can climb at is entirely dependent on excess thrust.  With props, maximum thrust can be at a lower airspeed than Vbg, so even if the plane is not at its optimal wing AOA (more drag), the maximum line between drag and thrust is where you get your best climb angle, Vx.
That's the easy part.  How can a plane climb faster when it is climbing at a lower angle?  By going faster!
The formula for Power is Force (thrust) x Velocity
When you see this graph, note that the maximum distance between Power and Drag is Vy.  This plane will fly higher and farther for a given amount of fuel than at Vx.