Why does Tarkovsky use monochrome and color in the same film?

I can't citate because I'm unsure where I read it, but one motivation for this is kind of obvious. Stalker is the main character so we mostly see everything the way he feels about it. This means that everything outside of Zone is grey and depressing for him. As soon as he is back in the Zone everything is colorful again and he feels alive. He even rolls on grass as a way to say hello to Zone that he missed so much.

He was in prison for being Stalker, they are being shot at when they try to enter the zone, but he still does it for some money. I think we can assume that father and husband would normally opt for some steadier income, not to mention one that is far less dangerous. In the end we conclude that it is him who is high on the Zone, or that Zone is part of him. In the end he tries to focus on his family although world is still grey and brown. That is Tarkovsky's way: even though we have fantastic sci-fi surrounding, the focus on the movie is on inner character's motives and development. In case of this movie we see this in all three Zone visitors, but only Stalker is struggling with the zone itself.

Tarkovsky, at least in the 60s, thought that color in film was a problem in that he seemed to feel it was pandering and commercially oriented and was some form of imposition on the audience.

That said, he would frequently use both black and white and color and (if I recall correctly) he even went as far as to have a scene in B&W with people watching a screen that had color footage playing on it.

Here is a quote from an interview he did in 1966:

On the screen colour imposes itself on you, whereas in real life that only happens at odd moments, so it's not right for the audience to be constantly aware of colour. Isolated details can be in colour if that is what corresponds to the state of the character on the screen. In real life the line that separates unawareness of colour from the moment when you start to notice it is quite imperceptible. Our unbroken, evenly paced flow of attention will suddenly be concentrated on some specific detail. A similar effect is achieved in a film when coloured shots are inserted into black-and-white.

(source)

While I feel the other answers are correct about potential symbolic meanings, he is also very much interested in the subjectivity of the moment.

Because it looks good. And indeed it does. Having the zone shown in color is to add a dramatic visual effect.