To explain this devotion we quote from texts in the public domain as follows.
The rose is blessed on the fourth Sunday of Lent, Lætare Sunday (hence also known as Rose Sunday), when rose-coloured vestments and draperies are substituted for the penitential purple, symbolizing hope and joy in the midst of Lenten solemnity. Throughout most of Lent, Catholics pray, fast, perform penance, and meditate upon the malice of sin and the terrible punishment it brings; Rose Sunday is an opportunity to look beyond Christ's death at Calvary and see Christ, the redeemer, risen in the first rays of the Easter sun, and rejoice. The shining golden flower shows forth Christ's majesty, appropriate because prophets called him "the flower of the field and the lily of the valleys." Its fragrance, according to Pope Leo XIII "shows the sweet odor of Christ which should be widely diffused by His faithful followers" (Acta, vol. VI, 104), and the thorns and red tint refer to His Passion. See Isaiah 63:2: "Why then is thy apparel red, and thy garments like theirs that tread in the winepress?"
The earliest roses were not blessed; instead, blessing was introduced to render the ceremony more solemn and induce greater reverence for it on the part of the recipient. According to Cardinal Petra (Comment. in Constit. Apostolicas, III, 2, col. 1), Pope Innocent IV (1245–54) was the first to bless it. However, others claim that Pope Innocent III (1198–1216), Pope Alexander III (1159–81) or Pope Leo IX (1049–55) was the first. It is said that Leo IX, in 1051, obliged the monastery (nuns) of Bamberg in Franconia, to furnish a Golden Rose to be blessed and carried on Lætare Sunday each year (Theop. Raynaud, De rosa mediana a pontifice consecrata, IV, 413). Pope Benedict XIV attests that the ceremony of blessing originated at the end of the 14th or the beginning of the 15th century. Catalanus, papal master of ceremonies, believes that even the earliest roses were anointed with musk and balsam, but the blessing with prayers, incense, and holy water had its inception later on, sometime before pontificate of Pope Julius II (1503–13). Currently, the pope blesses the rose every year, but it is not always a new and different rose; the old one is used until it has been given away.
Originally (before the papacy moved to Avignon) the rose was blessed in the Hall of Vestments (sacristy) in the palace where the pope was; but the solemn Mass and the donation of the rose took place in the Santa Croce in Gerusalemme (a figure, according to Pope Innocent III, of the heavenly Jerusalem). The blessing was followed by a solemn Mass sung either by the pope himself or the first Cardinal Priest. In the former case the rose was placed on a veil of rose-colored silk richly embroidered with gold; in the latter the pope held the rose in his hand, except while kneeling, or during the Introit, Confiteor, Elevation and the singing of "Laudemus in Domino". Rose in hand, the pope returned processionally to the Lateran Palace; the Prefect of Rome led his horse by the bridle and aided him in dismounting. Upon arrival, he gave the rose to the Prefect, as a recompense for these acts of respect and homage. Prior to 1305, the rose was given in Rome to no foreigner, except the Emperor on the day of his coronation. While residing at Avignon (1305–1375), the popes, unable to visit Roman churches and basilicas, performed many of their sacred functions, among them the blessing of the rose, in the private chapel of their palace (whence the origin of the Cappella Pontificia). On their return to Rome they (Sixtus V excepted) retained this custom.
The blessing of the rose now takes place in the Hall of Vestments (camera dei parimenti), and the solemn Mass in the papal chapel. The rose is placed on a table with lighted candles, and the pope, vested in alb and rose-colored stole and cope with precious mitre on his head, begins the ceremony with the usual versicles and the following poetical prayer:
"O God! by Whose word and power all things have been created, by Whose will all things are directed, we humbly beseech Thy Majesty, Who art the joy and gladness of all the faithful, that Thou wouldst deign in Thy fatherly love to bless and sanctify this rose, most delightful in odour and appearance, which we this day carry in sign of spiritual joy, in order that the people consecrated by Thee and delivered from the yoke of Babylonian slavery through the favour of Thine only-begotten Son, Who is the glory and exultation of the people of Israel and of that Jerusalem which is our Heavenly mother, may with sincere hearts show forth their joy. Wherefore, O Lord, on this day, when the Church exults in Thy name and manifests her joy by this sign [the rose], confer upon us through her true and perfect joy and accepting her devotion of today; do Thou remit sin, strengthen faith, increase piety, protect her in Thy mercy, drive away all things adverse to her and make her ways safe and prosperous, so that Thy Church, as the fruit of good works, may unite in giving forth the perfume of the ointment of that flower sprung from the root of Jesse and which is the mystical flower of the field and lily of the valleys, and remain happy without end in eternal glory together with all the saints."
The prayer finished, the pope puts incense (handed by the cardinal-deacon) into the censer and incenses the balsam and then the musk, and afterwards puts the balsam and powdered musk into the tiny cup in the heart of the principal rose. He then incenses the rose and sprinkles it with holy water. It is then given to the youngest cleric of the Camera, who carries it in front of the pope to the chapel, where it is placed on the altar at the foot of the cross upon a richly embroidered silk veil, where it remains during the Mass sung by the first cardinal-priest. After the Mass, the rose is carried in procession before the pope to the sacristy, where it is carefully put away in a place set apart for it, until bestowed upon some worthy personage.
Zenit reports on the use of the Papal Golden Rose by Pope Benedict XVI as follows...
The Golden Rose is a papal decoration conferred on prominent Catholic personalities; it has gone through a significant evolution. Initially, kings and dignitaries received it, later it was conferred almost exclusively on queens and, more recently, on Our Lady. The distinction was created by Pope Leo IX in 1049.
In more recent times, after the Second Vatican Council, the papal decoration has become almost exclusively a gift from popes to Our Lady. This was the 11th Golden Rose that the Pontiff has given to Our Lady in the more than five years of his pontificate. The other 10 were given to the Shrine of Jasna Góra in Poland (2006), the Basilica of Aparecida in Brazil (2007), the Mariazell Basilica in Austria (2007), the Basilica of the National Shrine of the Immaculate Conception in Washington, D.C. (2008), Our Lady of Bonaria in Cagliari, Italy (2008), Our Lady of Pompeii, Italy (2008), Our Lady of Europe in Gibraltar (2009), and the "Virgen de la Cabeza" (literally, Virgin of the Head) of the Diocese of Jaen, Spain (2009), the Shrine of Our Lady of Ta' Pinu (2010), and Our Lady of Fatima (2010).