In laymans terms, the radio wave in reality is actually 1/4 the size of it's wavethength in physical dimensions. When a radio wave hits a receiver's antenna, it induces a very tiny current flow in the antenna. If the antenna isn't long enough, part of the wave that hit the antenna is going to "miss", losing part of the signal. The missing portion of other waves could fill in, but that won't produce as good a signal.
Transmitters have a similar problem. When you put a signal into the antenna, it quite literally zooms up and back down the antenna emitting a radio wave as it goes. Here the antenna should be a multiple of 1/4 of the wavelength or you get part of the signal that doesn't get out of the antenna. Since it can't get out of the antenna, it goes right back into your transmitter. This is called a Standing Wave. In a tiny low power transmitter it is of little consequence. In a larger high power transmitter, you would soon be in need of a new transmitter as this is going to fry your final RF amplifier stage.
What they mean by the same "plane" is that the antenna is either pointing up and down or side to side in the same way as the transmitter's antenna. The radio waves actually orient themselves up and down or side to side. In fact, it is possible to receive two separate signals transmitted at the exact same frequency with two antennas. One mounted up and down, the other side to side. They call this polarization and it is a great way to cram more signal into the same bandwidth. It is even possible to have a circular polarization with clockwise and counterclockwise signals.