How does a color television function?

That is a huge question to be answered in one page. It requires understand of many scientific principles. Electronics, radio propagation, light and computer technology (these days).

Taking your question at it's most simplest:-

A TV receiver picks up a radio signal, that has been encoded and transmitted from a nearby station.

The signal is decoded inside the TV and a continuous stream of data is sent to the screen, line by line.

The picture is scanned across the screen from left to right, from top to bottom, making up a picture. Because your eyes average out fast changes in light, you don't see this line being drawn.

Each picture (or frame) is redrawn again from the top down. Again you don't see this, but you do see the differences in each picture, giving motion to the scene.

Early TVs used an electron beam fired at a phosphor coated glass screen, to make the picture. Modern TVs use a matrix of cells in liquid crystal or plasma cells, which are individually affected by addressing from a computer within the TV.