A polished apologetic formula combining several layers. 忙しさ nominalizes the adjective with ~さ. 紛れて ('lost in, swept up in') frames busyness as the distraction. ごぶさたする is a set phrase for 'being out of touch,' and しておりました layers the humble auxiliary おる (humble いる) over ~ている in polite past form — very deferential. This is the kind of greeting you'd use when reaching out to a superior or client after a long silence.