Where is the quote from? Gravity exerts a force, therefore by newton's second law there is acceleration. Nothing says the acceleration is purely angular. The closer the pass to the planet, the greater the acceleration.
Wiki explains it well. From the frame of reference of ship and planet, the relative speed is the same. But relative to the sun, the ship can be moving faster.
"A gravity assist around a planet changes a spacecraft's velocity (relative to the Sun) by entering and leaving the gravitational sphere of influence of a planet. The sum of the kinetic energies of both bodies remains constant (see elastic collision). A slingshot maneuver can therefore be used to change the spaceship's trajectory and speed relative to the Sun"
Wiki explains it well. From the frame of reference of ship and planet, the relative speed is the same. But relative to the sun, the ship can be moving faster.
"A gravity assist around a planet changes a spacecraft's velocity (relative to the Sun) by entering and leaving the gravitational sphere of influence of a planet. The sum of the kinetic energies of both bodies remains constant (see elastic collision). A slingshot maneuver can therefore be used to change the spaceship's trajectory and speed relative to the Sun"
https://en.m.wikipedia.org/wiki/Gravity_assist