@Hobbes' answer is correct about the what, but not the why.
Radio waves -- just like light -- are electromagnetic waves. Because they are transverse waves of the electric and magnetic fields, they can be polarized. There are two ways (*1) to polarize E-M waves: linear polarization and circular polarization.
- The electric field vibrates in one direction perpendicular to the direction of propagation, but with an amplitude that fluctuates periodically.
- Light example: polarized sunglasses
- Radio examples: dipole antennas; television, radio, Wifi, Apollo high-gain antenna
- The electric field maintains constant amplitude, but the direction changes as the wave propagates.
- Light example: polarized 3-D movie glasses
- Radio examples: spiral antennas, helical antennas, Apollo scimitar antenna
When E-M waves reflect off smooth surfaces, they can become linearly polarized. The best example of this is glare, which is light that reflects off a road, snow, or ground. In this case, the electric field perpendicular to the surface is reduced by destructive interference, and so we say the reflected light is polarized parallel to the surface. By wearing polarized sunglasses that are vertically polarized, you can block the glare, yet still see the ambient light.
The same destructive interference by reflection can also happen with radio waves. Television, radio, and WiFi are transmitted by dipole antennas that create a radio wave polarized in the direction of the antenna (vertical in the case of TV and radio). However, when the signal reflects off certain geographic features, you can get cancellation of the linearly-polarized signal. This is why some locations have a hard time getting TV or radio signals. However, this problem doesn't happen with circularly polarized signals.
Another issue with linear polarization is that when you rotate the transmitter (or receiver), the amplitude of the signal changes. If you tilt your head while wearing polarized sunglasses, the glare will start to appear. You could make a 3-D movie with linearly polarized glasses (one eye horizontally- and the other eye vertically-polarized), but you would see double-images if you tilt your head. It's also why the antenna on your WiFi router can be rotated into various orientations. This is another problem that doesn't happen with circularly polarized signals. Thus, 3-D movie glasses use circular polarization.
The disadvantage of spiral antennas is that they are inefficient. They tend to spread their signal in a wide beam, which means much of the energy of the radio transmission is being broadcast in the wrong direction. As @Hobbes notes, they can transmit on a wide bandwidth; you don't want to do this as it requires more transmitter power.
Putting a spiral or helical antenna on orbiting spacecraft is a smart idea. It creates circularly-polarized radio waves, which means the orientation of the antennas don't matter. That's important for an orbiting spacecraft, which is constantly changing its orientation relative to positions on Earth. There is also the advantage of circularly-polarized wave being less susceptible to interference by reflection. You can see a picture of a helical antenna on a ground-tracking station in my answer here.
In contrast, spacecraft beyond orbit -- think Apollo or Voyager -- have fairly stable orientations, and so they use more efficient linearly-polarized antennas. That's why Apollo had so many dang antennas: some for when the orientation was stable (*2), and others for when the orientation was changing (*3).
(*1) Elliptical polarization is a third way to polarize E-M waves, but they add nothing to this discussion.
(*2) CSM to Earth (trip between Earth and moon); LEM to Earth (on the moon); the S-IVB ("third stage") to Earth; and the ALSEP experiments left behind.
(*3) CSM to Earth (Earth or lunar orbit); LEM to CSM; LEM to Earth (ascending/descending); lunar rover to CSM.