The mitre is the single most legible object in Western liturgical dress, understood by people who could not name any other vestment, and it is far younger than the office it marks.
Worn, and taken off
A mitre in a church means a bishop is present and presiding. It is not worn continuously through a service: it goes on and comes off according to what the bishop is doing, off for prayer addressed to God, on for teaching and blessing, so the garment encodes the distinction between the bishop who prays with the church and the bishop who teaches it. Three grades, plain through jewelled, are chosen by the solemnity of the occasion.
A recent invention
Bishops had no distinctive headgear for the first thousand years of the church. The mitre appears only in the eleventh century, apparently as a papal grant to particular bishops before becoming general, which makes it a good illustration of how quickly an object can become so bound to an office that it looks immemorial. In Orthodoxy the corresponding headdress is a rounded crown following Byzantine imperial form rather than the Western cleft cap, a reminder that the shape is a cultural choice and not a necessity.