I remember hearing that asteroids and other objects can actually gain speed while passing a planet or large object to due the gravity, but still not enter orbit.
Is this true? If so, how is this possible? I'm asking because the phenomenon could be used to generate energy from gravity, and obviously we have not been able to do that.
It is true, and it works by transferring momentum from the large object to the small one. The movement of the large object is slowed to a degree equivalent to the energy transferred to the small object, but because usually the things involved are planets vs small asteroids, the change is imperceptible.
When used to increase the speed of space vessels, this is called a gravity assist or a slingshot maneuver. For instance, in the case of the Cassini probe, it has been used many times.
How you describe this interaction depends on your frame of reference.
If you look at things from the reference frame of the planet, then the object gains no energy or speed. The object approaches (speeding up) and departs (slowing down). The relative velocity between the object and the planet is the same at the same distance.
But we may instead choose to look at things from the reference frame of the sun. In that frame the momenta of the object and planet change. Some of the orbital momentum and kinetic energy of the planet is transferred to the object. There is no net increase or creation of energy in the interaction.
Velocity, momentum, and kinetic energy are all frame-dependent properties, and we sometimes choose different frames to examine things.
Is this true?
Yep! See Gravity assist. It's been used dozens of times, and some missions use many assists, sometimes from the same body more than once, and sometime to loose energy rather than gain it!
In these encounters, momentum and energy are still conserved. The spacecraft picks up speed by being pulled towards another much more massive body for a short time, usually hours.
If so, how is this possible? I'm asking because the phenomenon could be used to generate energy from gravity, and obviously we have not been able to do that.
Gravity goes both ways, and so the larger looses the same amount of energy as the spacecraft gains.
In the case of a spacecraft and a planet, the loss is so tiny that it is not measurable as a change in the planet's orbit, but energy is still conserved.
GIF (source)
You mention extracting power from gravity. We do that, for instance through tidal power stations or tide turbines. The tides are an effect of the Moon's gravity.
It would be possible in theory to do this in space -- you could launch a stream of magnetised objects so that they just reach the Moon's orbital distance, passing just behind the Moon in its orbit. The Moon's gravity will pull them forwards, and it should be possible to arrange this, so that they returns to Earth, but going faster than they were initially. This kinetic energy could be extracted by using it to induce current in a coil (which could be beamed down to Earth), slowing the objects back into their original orbits. It would be an incredibly elaborate and expensive way to extract a fairly small amount of energy though.
