What causes tangential velocity of a planet to change in an elliptical orbit?

Let go of a stone while high up in the sky.

• It comes crashing vertically down. Gravity pulls along with its velocity, so it speeds up and up and up.

• Repeat but this time push it a bit sideways also. It still comes crashing down but lands a bit further to the side. The sideways speed and the downwards speed caused by gravity combine into a slightly tilted velocity.

• Repeat but push much more sideways to start with. It now lands far away.

• Repeat again but now push so much that the object moves so much sideways that it misses Earth.

It will now fly past Earth. On the other side of Earth gravity pulls backwards in it, so it slows down and falls back again in the exact same way from the other side. Again it misses Earth. And this is repeated (ideally). This is an elliptic path.

• If you gave an even larger sideways speed to start with, the stone will miss Earth with a much larger margin, widening the ellisis.

• At some initial sideways speed the ellipsis is just as wide as it is heigh - it is now a circular path.

• An even larger sideways speed widens the circle so we again have an elliptic path, this time wide rather than tall.

In the circular motion the stone feels only a perpendicular pull from gravity. It's velocity is always perpendicular to the pull so it only feels a perpendicular acceleration $a=(0,a_perp)$ and thus it only turns. Without speeding up or slowing down.

In the vertical drop the stone feels only a parallel pull from gravity. It's velocity is always parallel/tangential to the pull so it only feels a parallel acceleration $a=(a_parallel,0)$ and thus it only speeds up/slows down.

In between these two scenarios we have velocities that are not perpendicular nor parallel to the pull from gravity. They are a bit of both. Thus the stone is both turning and speeding up/down simultaneously, $a=(a_parallel,a_perp)$. Each of these two acceleration components is created by the component of gravity that pulls perpendicular or parallel to the velocity because the velocity is pointing at an angle inside the gravitational field.