Subject: How does powered gravity assist work?

I've read on some webpages the following about powered gravity assist:
"A well-established way to get more energy from a slingshot is to fire a rocket
engine near the periapsis to increase the spacecraft's speed. Although a given
rocket burn will provide a the same change in velocity (delta-v) regardless of
when it occurs, the resulting change in kinetic energy is proportional to the
velocity. Therefore, to maximize the effect of the burn, gaining the most
kinetic energy for the propellant expended, the burn must occur at the moment
when the velocity is at its maximum, which occurs at periapsis." 
I understand the basic (non-powered) gravity assist affect and I also understand
that the energy increases more if the same fixes change of velocity is done at a
higher speed due the v square in the kinetic energy. From the above text I
undestand that the final velocity of the spacecraft after leaving a planet's
gravity well, is larger if if the same velocity change (delta-v engine burn) is
done at the periapsis (?closest to planet - largest velocity?) compared to any
other point along the spacecraft's trajectory (e.g, 'at the edge' of the
planet's gravity well. The burn is more efficient in generating more kinetic
energy at the point of highest velocity. Does this mean that the same amount of
fuel burned at a different velocity is generating more kinetic energy? This
somehow is not intuitive to me. 

Re: How does powered gravity assist work?