Keplers law says the planet sweeps out equal area slices of the ellipse-shaped pie in equal times. Thus when the planet is at its closest approach, aligned with the semi minor axis it has to be moving fastest in order to sweep out a big enough arc to obey the law. Moving fastest means it has the largest kinetic energy. Also it's the lowest potential energy. If you think of it like a pendulum, the closest approach of the planet is like the point in a pendulum's arc when it's straight down. It's moving fastest there and all potential energy has been converted to kinetic energy.
