The question has already been answered pretty well, but I think more explanation of why big antennas help would be worthwhile ...
Antennas have a property called "gain" which means that they work best in one direction (or several directions, depending on the type of antenna) and less well in others -- so a transmitting antenna with high gain sends its signal in one precise beam, and a receiving antenna with high gain receives best from one precise direction and mostly ignores signals from other directions.
Well, the gain of an antenna depends on the type of antenna it is and the ratio of its size to the wavelength of the signal. What this means for us is that if you want a really high gain antenna ... it has to be large compared to the wavelength of the signal.
The size of the antenna on the probe is fixed, but it's quite large, so it's got a high gain -- it sends its signal directly at where it knows the Earth is. On the Earth, we keep using bigger and bigger antennas to talk to the probe, so we get higher and higher gains, and a high gain antenna aimed properly picks up more of the signal and less noise.
(There are other factors involved, of course, but this is why the big antennas matter.)
Note that the higher the gain of your antenna, the more important it is that it is precisely aimed. But when we're talking to probes like this, we're aiming the antenna extremely precisely.
But for something like the WiFi in your house ... high gain antennas tend to be a bad thing rather than a good thing, because it means that the range is good (better than a lower gain antenna) in the given direction(s) but bad (worse than a lower gain antenna) in others.