Does the large physical size help the antenna radiate that much power?
When you're pumping out 100 KW of power, you want it all to go up the feedline and leave the antenna. Even a slight mismatch could result in melted cables! At 100 MHz, 1/2 wavelength is only about 1.5 meters. I don't want to be around when someone tries to slam 100 KW through a 1.5 meter long dipole!
Ed
Many people get confused by antenna gain.
A moment's thought will make it obvious that the only real power you have is that which your transmitter produces.
If 13dB of antenna gain, gave you a real power increase of 20times, you would have an "over-unity"device, & be able to solve the Earth's energy problems in one stroke!
Antenna gain is a measure of how much power would need to be applied to an "Isotropic" antenna (effectively a point source) to produce the same field strength in the direction you are interested in.
Hams often quote gain w.r.t a dipole---- to convert that to EIRP add approx 2dB to the quoted figure.
From the above, you can see that all your antenna system has to handle is the transmitter output, in my quoted case, 5kW, so the requirements aren't as severe as you believed.
Furthermore, as 4CX35000 points out, there are multiple dipoles, each of which only need be rated for a fraction of the transmitter output power.
The large physical size comes down to several things:-
Thick elements are more wide banded than thin ones.
They are also more rugged, which is important in antennas which have to sit on top of high towers in all sorts of weather, for decades.