The modern manager, buried in spreadsheets and strategy meetings, descends from a horse trainer. Italian maneggiare meant 'to handle, to work with the hands, to train horses', from Latin manus ('hand'). To manage was to control a horse through the hands on the reins.
The word entered English in the 1560s, initially in its equestrian sense. A manège (still used in dressage) is a riding school or training arena. The leap from horse training to business administration happened quickly: by the 17th century, manage meant directing any enterprise, not just a stable.
Latin manus is one of the most productive roots in English. Manual: done by hand. Manuscript: written by hand. Manufacture: originally made by hand (now ironic, given factories). Manoeuvre: working by hand, from French main ('hand') + oeuvre ('work'). Manipulate: to handle skillfully, from manipulus ('handful'). Manner: the way something is handled. Manacle: a hand-chain
The 'cope with difficulty' sense — 'I'll manage' — appeared by the 18th century. It carries an echo of the original horsemanship: managing a difficult horse required patience and skill under pressure. To manage a crisis is to keep your hands steady on the reins when the horse wants to bolt.
Spanish took the same root in a different direction: manejar means 'to drive a car'. The hand still steers.