There are two forms of this one-armed freestyle, depending on which side you breath on. Supposedly each one highlights some aspect of balance and timing, but I won't try to get in to that, because I'm only familiar with the version I use, where I breath on the side of the stroking arm. I also like to swim this way with my hand closed in a fist. I can slow the stroke way down when I do this, which allows me to focus on my timing of hips and shoulders. The other thing this seems to help is precisely the bobbing that ZT was referring to. A slow one armed recovery means that your recovering arm is spending more time out of the water, which pushes your body down. This can help you to time your breathing with the natural bobbing of your body induced by the heavy arm outside the water. At more realistic swimming rates, this effect becomes smaller, so it is harder to teach yourself how to exploit it and coordinate it with your breathing.
Some years ago there was a long thread on this forum about one-armed breathing, which I can't find now, but others with better web skills might be able to put up a link to it. Two coaches who saw some virtue in this drill were Charles Couterier and Suzanne Atkinson.