While this is probably not the sole reason Stephan Connerman mentions (in "Der Indische Ozean in historischer Perspektive") that arab/muslim merchants settled, intermarried with the east african population and finaly took controle of some coastal regions as part of their mercantile exploits in the indian ocean. This would point to trade, or rather the further inclusion of east africa into the indian ocean trade network, as one reason for the spread of Islam in that region.

What gave you the impression it "spread so readily"? As far as I can tell, the sources we know are pretty scant and don't imply some rapid proselytization. It could have taken centuries