Ok, so please just tell me why the length of an one-pole Antenna should be 1/4 of the signal wavelength so that the maximum power is sent?
There is a lot of scope for misunderstanding in that statement!
"One-pole antenna"
All antennas must have two poles, the second pole may not be obvious, but it will be there or else it would not work at all.
I assume you mean a single vertical wire or rod.
That single wire should be above a conductive ground plane for it to work correctly.
Many radios have just a single antenna and rely on the rest of the circuit of the radio to provide the ground plane. Not very efficient but it does work.
"1/4 of the signal wavelength so that the maximum power is sent"
Lots of possible variables here.
Depending on the length of the antenna as a number of wavelengths at the operating frequency, so the radiation pattern of the antenna varies, it puts more signal in one direction than another.
Also, as the length and configuration of the antenna varies, so does its feedpoint impedance.
A quarter wave monopole above a ground plane has a feed impedance of about 35 ohms.
A halfwave dipole has a feed impedance of about 72 ohms.
A folded dipole has a feed impedance of about 300 ohms.
The transmitter (a real transmitter, not the kiddie FM noise toys so often discussed here on ETO) will be designed to produce its rated output power into some defined load, usually 50 ohms.
If the transmitter is connected to a load other than what it was designed for, it will not produce its rated output power.
A matching network is used to transform the antenna feed impedance to that which the transmitter is expecting.
So going back to your question:
why the length of an one-pole Antenna should be 1/4 of the signal wavelength so that the maximum power is sent?
There is not a simple answer, and depending on the configuration of the various parts of the system, a quarter wavelength antenna may not be the one which transmits maximum power.
JimB