Literally, to being in or introduce. Hence, appropriately,

To introduce, as to a benefice or office; to put in actual possession of an ecclesiastical living or of any other office, with the customary forms and ceremonies. Clerks or parsons are inducted by a mandate from the bishop to the archdeacon, who usually issued a precept to other clergymen to perform the duty. In the United States, certain civil officers and presidents of colleges, are inducted into office with appropriate ceremonies.