The state of being taken up with something. Also the measured attention a thing attracts — and the second sense is where the first gets exploited.
Old French engagier, en- (to put into) plus gage (a pledge), of Frankish origin; a taking-up bound by a pledge.
Its root is a pledge, not a glance: to be engaged was to be bound by something staked. The measured, tracked kind of attention this word now also names is exactly where that old binding gets turned against the one bound.