The Church permits cremation, provided that it does not demonstrate a denial of faith in the resurrection of the body. (n. 2301)
The above quotation is footnoted with a reference to Canon 1176 #3 which states:
The Church earnestly recommends that the pious custom of burying the bodies of the deceased be observed; nevertheless, the Church does not prohibit cremation unless it was chosen for reasons contrary to Christian doctrine.
Given this grudging permission ("does not prohibit") how is it that cremation has become so common amongst Catholics? It has always seemed to me unworthy of the dignity of the body of a baptised person - being a Temple of God - to consign it to the flames of a cremator. As Christ himself descended into the tomb after His death in anticipation of His Resurrection, so burial is a more perfect imitation of Christ the Lord as our bodies rest in the tomb awaiting the resurrection of the dead at the Last Day.
I have just finished reading John Henry Newman's "Development of Christian Doctrine". He quotes an early witness of the Christians who testifies that "They execrate the funeral-pile." That is to say: they do not burn the bodies of their deceased but rather bury them piously. Pagans, on the other hand, "hold corpses and sepulchres in aversion." For pagans, because matter, and therefore the human body, is essentially bad, it can be burned. Yet for Christians, the body is a holy thing having been united to the Divine Person of the Son of God.
And of course Catholic venerate the relics of the saints, their very bodies.
Here is the relevant section of Newman's work about this matter which you can read in its full context here:
"Perish the thought," says Manes, "that our Lord Jesus Christ should have descended through the womb of a woman." "He descended," says Marcion, "but without touching her or taking aught from her." "Through her, not of her," said another. "It is absurd to assert," says a disciple of Bardesanes, "that this flesh in which we are imprisoned shall rise again, for it is well called a burden, a tomb, and a chain." "They execrate the funeral-pile," says Cæcilius, speaking of Christians, "as if bodies, though withdrawn from the flames, did not all resolve into dust by years, whether beasts tear, or sea swallows, or earth covers, or flame wastes." According to the old Paganism, both the educated and vulgar held corpses and sepulchres in aversion. They quickly rid themselves of the remains even of their friends, thinking their presence a pollution, and felt the same terror even of burying-places which assails the ignorant and superstitious now. It is recorded of Hannibal that, on his return to the African coast from Italy, he changed his landing-place to avoid a ruined sepulchre. "May the god who passes between heaven and hell," says Apuleius in his Apology, "present to thy eyes, O Emilian, all that haunts the night, all that alarms in burying-places, all that terrifies in tombs." George of Cappadocia could not direct a more bitter taunt against the Alexandrian Pagans than to call the temple of Serapis a sepulchre. The case had been the same even among the Jews; the Rabbins taught, that even the corpses of holy men "did but serve to diffuse infection and defilement." "When deaths were Judaical," says the writer who goes under the name of St. Basil, "corpses were an abomination; when death is for Christ, the relics of Saints are precious. It was anciently said to the Priests and the Nazarites, 'If any one shall touch a corpse, he shall be unclean till evening, and he shall wash his garment;' now, on the contrary, if any one shall touch a Martyr's bones, by reason of the grace dwelling in the body, he receives some participation of his sanctity." Nay, Christianity taught a reverence for the bodies even of heathen. The care of the dead is one of the praises which, as we have seen above, is extorted in their favour from the Emperor Julian; and it was exemplified during the mortality which spread through the Roman world in the time of St. Cyprian. "They did good," says Pontius of the Christians of Carthage, "in the profusion of exuberant works to all, and not only to the household of faith. They did somewhat more than is recorded of the incomparable benevolence of Tobias. The slain of the king and the outcasts, whom Tobias gathered together, were of his own kin only."