This is because it has been awarded more indulgences than any other confraternity in the Church, and ever since its inception there has hardly been a Pope who has not opened up the Treasures of the Church to enrich it with further privileges.
Knowing that a good example is more compelling than glowing words and even favors, Sovereign Pontiffs have found that there was no better way to show their high regard for the confraternity than to join it themselves.
Here is a short summary of the indulgences which they wholeheartedly granted to the Confraternity of the Holy Rosary and which were confirmed again by our Holy Father Pope Innocent XI on July 31, 1679 and received and made public on September 25th of the same year by His Excellency the Archbishop of Paris:
1. Members may gain a plenary indulgence on the day of joining the confraternity;
2. A plenary indulgence at the hour of death;
3. For each three groups of Five Mysteries recited: ten years and ten quarantines;
4. Each time that members say the holy names of Jesus and Mary devoutly: seven days' indulgences;
5. Seven years and seven quarantines may be gained by those who devoutly take part in or attend the Holy Rosary Procession;
6. Members who have made a good confession and who are genuinely sorry for their sins may gain a plenary indulgence on certain days by visiting the Holy Rosary Chapel in the Church where the Confraternity is established. This plenary indulgence can be gained on the First Sunday of every month, and on the feasts of Our Lord and Our Lady.
7. For assisting at the Salve Regina  one hundred days' indulgence;
8. Those who openly wear the Holy Rosary out of devotion and to set a good example may gain one hundred days' indulgence;
9. Sick members who are not able to go to Church may gain a plenary indulgence by going to confession and receiving Holy Communion and by saying that day the whole Rosary if possible, or at least five decades;
10. Our Sovereign Pontiffs have shown their generosity towards members of the Rosary Confraternity by allowing them to gain the indulgences attached to the Stations of Rome by visiting five altars in the Church where the Rosary Confraternity is established, and by saying the Our Father and Hail Mary five times before each altar, for the happy estate of the Church. If there are only one or two altars in the Confraternity Church they should recite the Our Father and Hail Mary twenty-five times before one of them.
This is a wonderful favor granted to Confraternity members for in the Stational Churches in Rome plenary indulgences can be gained, souls can be delivered from Purgatory and many other great indulgences too can be gained by members with very little effort and no expense and without leaving their own country. And even if the Confraternity is not established in the place where the members live they can gain the very same indulgences by visiting five altars in any Church. This concession was granted by Leo X.
The Sacred Congregation of Indulgences drew up a list of certain definite days upon which those outside the city of Rome could gain the Indulgences of the Stations of Rome. The Holy Father approved this list on March 7th, 1678, and commanded that it be strictly observed. These indulgences can be gained on the following days:
All the Sundays of Advent; each of the three Ember Days; also Christmas Eve, at Midnight Mass, the Daybreak Mass and at the Third Mass; the feast of Saint Stephen; that of Saint John the Evangelist; the feast of the Holy Innocents; the Circumcision and the Epiphany; the Sundays of Septuagesima, Sexagesima, Quinquagesima and on every single day from Ash Wednesday to Low Sunday inclusively; each of the three Rogation Days; Ascension Day; the Vigil of Pentecost; every day during the octave; and on each of the three September Ember Days.
Dear Confraternity members, there are numerous other indulgences which you can gain. If you want to know about them look up the complete list of indulgences which have been granted to members of the Rosary Confraternity. You will see the names of the Popes in question, the years in which they granted the indulgences and many other particulars which I have not been able to include in this little summary.