The meditation anchor works in a similar manner to a ship anchor. A ship can be at sea for days, being rocked and turned by the waves, but when a sailor drops an anchor, the boat remains put, not pitching about or running aground.
We use an anchor in meditation to protect our minds from being knocked about by a sea of feelings, stimuli, experiences, sounds, and emotions.
There are a number of apparent reasons to use our breath as a source of stability. To begin with, it is still with us. One thing we can count on when we are alive is that we will be breathing. Second, it is fairly open and easy to feel for the majority of people.
The majority of novice meditators will easily add mindfulness to their own breathing. Third, the breath is normally neutral—we don't have strong emotions about it.
Breath is a great meditation object because it is neither enjoyable nor bad, because if we choose an anchor that evokes intense negative or positive emotions, we will spend more of our time manipulating our feelings rather than being present with the breath.
You may also want to read more about Mindfulness Meditation and Healing here.