CRT televisions have seperate video amps for the red, blue and green "guns" in the CRT. There are several ways of driving these circuits (I-Q, R-Y & B-Y, XZ among others) but they all perform the same basic function. These circuits can be checked fairly easily as can the preceding chroma and video circuits. Alas, the filament in the CRT can go but this is easily checked with a resistance measurement (if you have a schematic or can otherwise determine the CRT's pinout.) Sometimes, if the CRT "gun" is gone, it will be impossible to display a true monochrome picture, though this is not a fool proof troubleshooting method.
However, if you have to ask "can this be fixed", you probably do not have the background knowledge to repair it. You probably don't have the equipment either! There are web sites with detailed troubleshooting and repair tips and pointers. I don't have any links on this PC; try Googling "sam goldwasser" or "TV repair faq". Sam posted a lot of his electronic repair writing on the usenet and it has been ported over to the internet.
As for explaining how TV's work - well that's more than an internet site can cover in detail. Personally, I have many books on TV and other electronic repair - it's been a hobby of mine for 35-40 years. Alas, today's economy deals with disposable consumer products but I suspect you can still find up to date books on electronic repair - try Sams (no relation to goldwasser above) or Wiley & sons.