A simplified diagram of the circuit is shown below. It divides the input from the source down two conductors that are each cut to be exactly as long as one quarter of the wavelength of the microwave signal supplied by the source. The power is automatically divided due to one of the properties that physicists love: symmetry. Faced with no difference in the two paths it's presented with, the input microwave signal splits and half of its power travels down each path. That's kind of cool, but if that's all that Wilkinson intended, he could have used any conductor length.
The quarter wavelength conductors provide an ingenious way of protecting each supplied circuit from the
other ones. Under normal operation, there's no potential difference at the end of the divider where the resistor connects the two conductors, so no current flows through the resistor. If one of the supplied circuits begins to force reflected power back into the divider, the resistor comes into play. It immediately shunts part of the reflected power over to the other output. That sounds bad, but wait for it.
The portion of the reflected signal, (step 1 in picture 3) that isn't shunted will travel back down the quarter wavelength conductor changing it's phase by 90 degrees, (a quarter wavelength), in the process, (step 2 in picture 3). At the top of the divider it will travel the opposite direction down the other conductor and in so doing pick up another 90 degree phase shift. At this point, it's picked up 180 degree phase shift and perfectly interferes with itself in the second conductor cancelling itself out! (step 3 in picture 3)
1. Original circuit design
Wilkinson E.J. (1960). An N-Way Hybrid Power Divider, IEEE Transactions on Microwave Theory and Techniques, 8 (1) 116-118. DOI: 10.1109/TMTT.1960.1124668