The flyback circuit is a classic method of generating moderately high voltages. It is not, by any means, the ONLY way, but it was once a very common circuit, and was employed in EVERY CRT based TV or monitor.
It is a VERY clever circuit, because it solves two design problems in the system with one circuit.
The scanning electron beam in a CRT needs to be a.) created and b.) scanned.
All pictures bar two are taken directly from Wikipedia.
Step 1: Creating an electron beam
SO, we need a high voltage.
Step 2: Scanning the beam
Well, an electron beam is a current, a current passing through a magnetic field experiences a force at right angles to the direction of both the field and the current.
So if we create a field, at right angles to the beam, we can cause the beam to bend, and scan.
If a current passes through a coil, it creates a field, if the current increases linearly, the field follows, if the field follows, the bending in the electron beam follows linearly too.
....we need a linearly ramping current
Oh, and at the END of the scan, it would be really nice if the trace flew back VERY quickly, so fast you can't see it happen. That's the "Flyback"
Wikipedia: Raster scanning