Assuming that you meant “But what is the antenna theory”, it’s a transformer with a specific turns ratio between primary and secondary. From the turns ratio an impedance transformation can be derived. The impedance ratio is the square of the turns ratio. The purpose of the ferrites is to increase the inductance of the wires passing through the cores, to allow them to operate as a transformer at a lower frequency than would be possible if the cores were say, cardboard or plastic. Using ferrite cores allows for reduced number of turns, which means lower losses. However ferrite cores have limits to their power handling and also they have ideal frequency ranges; outside of their ideal range the inductance doesn’t have the same increase.
Usage with an antenna. If the antenna impedance is not close to the impedance of the feedline in use, some means of transforming the impedance of the antenna to that of the feedline must be used. For example a high impedance antenna of 2500 ohms could be fed by 50 ohm feedline if you had a transformer with an impedance multiplication of 50. 50 is not a square of a simple turns ratio. The nearest perfect square you can find is 49 which would use a turns ratio of 7.
Tuned circuits can be used for changing impedance too, in an adjustable antenna matchbox or antenna tuning unit. The antenna matchbox has the slight disadvantage of requiring tuning for each specific frequency or band used. It doesn’t tune the antenna, it transforms the impedance of the antenna so it looks as if it is 50 ohm.
Broadband transformers like this can transform the load impedance to one that is usable with typical feedline/feeder and operate without adjustment over a range of frequencies and bands. A typical frequency range for such devices is 3 to 20 mhz, 3 to 30 or 2 to 10 MHz depending on the cores used, the number of turns, thickness of wire, configuration used. If increased losses are tolerable for the convenience of a no-tune device, wider frequency ranges are possible.
Entire text books have been written on this subject. I hope this description provides the basics.
No doubt some of the finer points are missing from my description will make the purist weep. But I think this is adequate for practical applications.
Andrew