Acceleration on approach and slowing on departure
A spacecraft accelerates as it falls toward a planet, reaches a greater speed near closest approach and slows while receding. In an ideal unpowered encounter with other perturbations neglected, incoming and outgoing asymptotic planet-relative speeds match, but their directions differ.
This compares far-field speeds rather than claiming constant speed throughout the encounter. Gravitational acceleration remains present; a freely falling vehicle can feel weightless while its trajectory bends.
Why another frame shows an energy change
The planet also moves around the Sun. Adding its heliocentric velocity to each planet-relative spacecraft vector can produce heliocentric speeds of different magnitudes, because the encounter rotates the relative vector.
When the spacecraft gains heliocentric orbital energy, the planet loses a minute corresponding amount. The spacecraft’s much smaller mass permits a substantial change for it and a tiny reaction on the planet, consistent with conservation.
Assists can increase or reduce energy
Encounter geometry can raise or lower orbital energy; merely approaching a planet does not guarantee an outward boost. NASA illustrates outward assists with Voyager and an energy-reducing encounter with Galileo passing Io during Jupiter arrival.
A mission must arrange encounter time, direction and distance and control trajectory errors. Assists can reduce propulsion requirements without replacing every launch, correction or insertion maneuver, and choosing a flyby changes the route and schedule.