I think in the book he gives a good description of a child refusing to have their nails cut. I can't find whereabouts it is though. But basically, he described this issue (and I'm guessing this applies to tooth-brushing issues too) as being one of a power imbalance. You have to create a setting in which your child feels that they are in control of the toothbrushing. He suggests playing around with this notion by approaching and backing away, e.g. by negotiating just to brush one tooth, and then stopping as soon as the child says 'Stop!' and backing away, then negotiating coming closer, backing away, closer, backing away etc etc... perhaps not even actually brushing any teeth initially. The purpose of this is to enable your child to feel that they are in control of the situation, and that you are not going to force them to do anything that they don't want to do. So by stopping when they ask you to stop, you are building up trust (or rebuilding it if you've had a tussle over this in the past).
This scenario needs to be played out like a game, with no pressure, no coercion, etc. It sounds time consuming - and it probably will be initially. But after two or three attempts at this game, your child will probably feel more relaxed about the whole thing, will trust you more, and will have clean teeth!
Using puppets, letting them brush your teeth as well, is also a good idea.
Also, reversing roles can be really helpful. Say you have really grimy teeth, you really need to brush them, but don't know how to, and you really don't want to. The aim here is to get them to help you, but you have to really exaggerate resisting them, flail around in feigned agony (but make it obvious you're playing!) - this is likely to get them to adopt the role of encouraging you, comforting you, and also perhaps acting out some of their fears around toothbrushing. E.g. if they respond to your feigned fear by being forceful, let them do it, and really act our the fear, let them see you know it's been scary for them - this is one of the fundamental elements of playful parenting. By observing you being scared, it helps them to process their fear.
Hope that makes sense. My description is a weak version of what you'd find in the book though, but hope it helps!