Say we have a Wilkinson power divider:
The explanation I've heard about how ports 2 and 3 are isolated is this: say there's a signal generator attached to port 2. This can take two paths to port 3: through the resistor, and through the two transmission lines. With the two transmission lines being a half wavelength, they invert the signal. So the non-inverted signal (through the resistor) and the inverted signal (through the transmission lines) is equal but opposite and so cancels.
That makes sense, but it's not enough to convince me that it actually works. For example, why split in half and not some other ratio which would not result in complete cancellation? And wouldn't port 1 have some significance to the operation?