This feast was first instituted by the Carmelites around 1380 to celebrate the approbation of thier name and constitution by Pope Honorius. The feast was assigned to July 16, because on that date in 1251, according to Carmelite tradition, the scapular was given by the Blessed Virgin to Saint Simon Stock. We will read from the Office of Readings of the old Carmelite Breviary:

When on the holy day of Pentecost, the Apostles, through heavenly inspiration spoke in diverse tongues and worked many miracles by the invocation of the most holy name of Jesus, it is said that many men who were walking in the footsteps of the prophets Elias and Eliseus, and had been prepared for the coming of the Christ by the preaching of John the Baptist, saw and acknowledged the truth, and at once embraced the faith of the Gospel. These new Christians were so happy to be able to enjoy the company of the Blessed Virgin Mary, and venerated her with so special an affection, that they, before all others, built a chapel to the purest of Virgins on that very spot of Carmel where Elias of old had seen the cloud, a remarkable type of the Virgin ascending.

Many times each day they came together to the new oratory, and with pious prayers, ceremonies, and praises, honored the Blessed Virgin Mary as the protector of their order. For this reason, people from all parts began to call them the Brethren of the Blessed Virgin of Mount Carmel; and the sovereign pontiffs not only confirmed this title, but granted special indulgences to whoever called either the whole Order or individual brothers by that name. But the most noble Virgin not only gave them her name and protection; she also bestowed upon blessed Simon Stock, the Englishman, the holy Scapular as a token, wishing the holy Order to be distinguished by that garment, and to be protected by it from the evils which were assiling it. Moreover, as formerly the Order was unknown in Europe, and on this account, many were importuning Pope Honorious for its abolition, the loving Virgin Mary appeared by night to Honorious, and clearly bade hiim receive both the order and its members with kindness.