Wavelength is inversely proportional to frequency, and directly proportional to the speed of the signal in the medium (think speed of light).
A half-wave antenna is thus a length equal to half the wavelength of the signal in the material the antenna is composed of. So your question should be "What is so special about a half wavelength as opposed to any other length" (not wavelength).
Imagine a wave machine at the end of a long narrow swimming pool. If the wave machine makes a single wave, it goes to the end of the pool, is reflected from the end wall and returns to the wave machine where it is reflected again, the wave going back and forth several times before it dies out..
Imagine the wave machine generating a series of waves. These waves will bounce off the end wall and return to the wave machine.
If the waves come back exactly so they match the wave being created by the wave machine then the reflection and the wave being created reinforce each other.
If the waves come back at some other point, then the waves will interfere, and the pattern of signal and reflection will become complex (I'll avoid the term chaotic)
The factors which determine whether the waves are timed so that they reinforce each other, or not are determined by the length of a wave (from crest to crest), the speed of the wave, and the length of the pool.
It happens that lengths of 1/4 wavelength, and 1/2 wavelength have interesting properties (if you consider what happens to the reflections -- 1/2 wavelength corresponds to a full wavelength by the time the reflection gets back to the source)
