I an not sure about the BMW-specific MAF, but many MAF sensors determine the air flow by heating a small wire or thin metal placed in the airstream. It then measures how much current is required to keep the wire at a constant temperature, e.g., the greater the mass of air flowing past the wire, the more current it takes to keep the wire hot. If it gets dirty, it insulates the MAF wire and the MAF then indicates a smaller mass of air flow going by the MAF than the actual amount of air mass going by the MAF. The computer then injects too little fuel, since the MAF has indicated less air was flowing than the true value. This results in too much air for the fuel injected and hence, "Too Lean."

As to why only one bank, well, in theory you should get an indication for both banks. But since an MAF problem due it it being dirty is typically a very slow process, gradually getting worse as time/miles pass, it is not unusual for the O2 sensor for one bank or the other to be the first to cross the threshold and indicate a "too lean" problem, e.g., one O2 sensor may simply be slightly different to the other O2 sensor due to manufacturing tolerances or aging, deposits, etc. So one of the O2 sensors will be the first to "pick it up," and the other will follow at some point down the road as the condition worsens.

Of course, it might also not be the MAF, and could be something specific to one bank or the other.