Pope Leo XIII is known as the "Pope of the Rosary". This is because, of the 81 encyclicals he wrote during his pontificate (1878-1903), twelve concern the Virgin Mary, and eleven the Rosary: these encyclicals form a series of teachings and exhortations, in which the Holy Father highlights the importance of this prayer.
Here are the historical reasons that led the Pope to make this tenth month of the year the "month of the Rosary", found in his Encyclical Supremi apostolatus officio of September 1, 1893:
Towards the end of the 12th century, Albigensian heretics, who sprung from the sect of the later Manicheans, and who filled the South of France and other portions of the Latin world with their pernicious errors, and carrying everywhere the terror of their arms, strove far and wide to rule by massacre and ruin.
Our merciful God, as you know, raised up against these most direful enemies a most holy man, the illustrious parent and founder of the Dominican Order. Great in the integrity of his doctrine, in his example of virtue, and by his apostolic labours, he proceeded undauntedly to attack the enemies of the Catholic Church, not by force of arms; but trusting wholly to that devotion which he was the first to institute under the name of the Holy Rosary, which was disseminated through the length and breadth of the earth by him and his pupils. Guided, in fact, by divine inspiration and grace, he foresaw that this devotion, like a most powerful warlike weapon, would be the means of putting the enemy to flight, and of confounding their audacity and mad impiety. Such was indeed its result. Thanks to this new method of prayer - when adopted and properly carried out as instituted by the Holy Father St. Dominic - piety, faith, and union began to return, and the projects and devices of the heretics to fall to pieces.
[...] Not only do We earnestly exhort all Christians to give themselves to the recital of the pious devotion of the Rosary publicly, or privately in their own house and family, and that unceasingly, but we also desire that the whole of the month of October in this year should be consecrated to the Holy Queen of the Rosary.
We decree and order that in the whole Catholic world, during this year, the devotion of the Rosary shall be solemnly celebrated by special and splendid services. From the first day of next October, therefore, until the second day of the November following, in every parish and, if the ecclesiastical authority deem it opportune and of use, in every chapel dedicated to the Blessed Virgin - let five decades of the Rosary be recited with the addition of the Litany of Loreto. We desire that the people should frequent these pious exercises; and We will that either Mass shall be said at the altar, or that the Blessed Sacrament shall be exposed to the adoration of the faithful, Benediction being afterwards given with the Sacred Host to the pious congregation.
Pope Leo XIII, Supremi Apostolatus Officio, September 1st, 1883