In order to understand this, it is necessary to understand the possible bases for Catholic a theological belief.
My answer here discusses the role of Sacred Tradition and Sacred Scripture in the Church; and my answer here begins a discussion of the Magisterium of the Church. These three are the grounds on which the Catholic Church can declare something to be a belief to be held by the entire Church.
Briefly, the Church holds Sacred Tradition to be passed down from the apostles, just as the scriptures were:
The Tradition here in question comes from the apostles and hands on what they received from Jesus' teaching and example and what they learned from the Holy Spirit.
(Catechism of the Catholic Church, paragraph 83)
The Magisterium is the authority the Church claims, which it holds to be passed down from the apostles, to interpret Scripture and Tradition so as to guide the Church unerringly in her beliefs:
The apostles entrusted the "Sacred deposit" of the faith (the depositum fidei), contained in Sacred Scripture and Tradition, to the whole of the Church. "By adhering to [this heritage] the entire holy people, united to its pastors, remains always faithful to the teaching of the apostles, to the brotherhood, to the breaking of bread and the prayers." ... "The task of giving an authentic interpretation of the Word of God, whether in its written form or in the form of Tradition, has been entrusted to the living, teaching office of the Church alone. Its authority in this matter is exercised in the name of Jesus Christ. This means that the task of interpretation has been entrusted to the bishops in communion with the successor of Peter, the Bishop of Rome.
(Catechism of the Catholic Church, paragraphs 84–85)
One thing I did not discuss in either of the above answers is what is called the sensus fidei, or "sense of the faithful". Under certain conditions, when it appears that all of the Church holds a particular belief, the Church may declare that the belief is a truth to be held as dogma:
All the faithful share in understanding and handing on revealed truth. They have received the anointing of the Holy Spirit, who instructs them and guides them into all truth."
"The whole body of the faithful... cannot err in matters of belief. This characteristic is shown in the supernatural appreciation of faith (sensus fidei) on the part of the whole people, when, 'from the bishops to the last of the faithful,' they manifest a universal consent in matters of faith and morals."
"By this appreciation of the faith, aroused and sustained by the Spirit of truth, the People of God, guided by the sacred teaching authority (Magisterium),... receives... the faith, once for all delivered to the saints.... The People unfailingly adheres to this faith, penetrates it more deeply with right judgment, and applies it more fully in daily life."
(Catechism of the Catholic Church, paragraphs 91–93; the quotes are from the Dogmatic Constitution Lumen Gentium.)
Thus, the Magisterium can be exercised by the People of God in its entirety (under certain conditions), by the entire College of Bishops, and by the Pope. When Pope Pius wrote Munificentissimus Deus, he invoked all three. A few years earlier, he had written a letter to the bishops titled Deiparae Virginis Mariae, in which he reviewed the fact that many people, religious and lay, had asked various Popes over the previous hundred years to make the belief in the Assumption into a matter of dogma. He asked the bishops what they and their dioceses thought:
We earnestly beg you to inform us about the devotion of your clergy and people (taking into account their faith and piety) toward the Assumption of the most Blessed Virgin Mary. More especially We wish to know if you, Venerable Brethren, with your learning and prudence consider that the bodily Assumption of the Immaculate Blessed Virgin can be proposed and defined as a dogma of faith, and whether in addition to your own wishes this is desired by your clergy and people.
Apparently, this was answered with a strong "Yes"; in Munificentissimus Deus the Pope reports:
those whom "the Holy Spirit has placed as bishops to rule the Church of God"(4) gave an almost unanimous affirmative response to both these questions. This "outstanding agreement of the Catholic prelates and the faithful,"(5) affirming that the bodily Assumption of God's Mother into heaven can be defined as a dogma of faith, since it shows us the concordant teaching of the Church's ordinary doctrinal authority and the concordant faith of the Christian people which the same doctrinal authority sustains and directs, thus by itself and in an entirely certain and infallible way, manifests this privilege as a truth revealed by God and contained in that divine deposit which Christ has delivered to his Spouse to be guarded faithfully and to be taught infallibly.
In other words, because essentially all the bishops and all the faithful agreed that it was the truth, the Pope declared that it must be truth (since the Church as a whole could not err in matters of faith), and should be taught infallibly, as a truth to which all the faithful must adhere.
Some of the support that Pope Pius gives comes from the celebration of the Dormition of Mary from ancient times; for example:
the fact that a holy fast had been ordered from ancient times for the day prior to the feast is made very evident by what our predecessor St. Nicholas I testifies in treating of the principal fasts which "the Holy Roman Church has observed for a long time, and still observes."
(Munificentissimus Deus, paragraph 19; citing the "Responsa Nicolai ad consulta Bulgarorum" ["Response of Nicholas to the actions of the Bulgars"], a letter sent by Pope St. Nicholas I to Boris, king of the Bulgars, in the late 9th century)
He further cites a sermon of St. John Damascene ("In Praise of the Dormition of Mary, the Mother of God [Theotokos] and Ever-Virgin"), which states:
It was fitting that she, who had kept her virginity intact in childbirth, should keep her own body free from all corruption even after death. It was fitting that she, who had carried the Creator as a child at her breast, should dwell in the divine tabernacles. It was fitting that the spouse, whom the Father had taken to himself, should live in the divine mansions.
After following this up with a discussion of how other saints and theologians have agreed with these ideas, Pope Pius concluded:
Since our Redeemer is the Son of Mary, he could not do otherwise, as the perfect observer of God's law, than to honor, not only his eternal Father, but also his most beloved Mother. And, since it was within his power to grant her this great honor, to preserve her from the corruption of the tomb, we must believe that he really acted in this way. ...
Since the universal Church, within which dwells the Spirit of Truth who infallibly directs it toward an ever more perfect knowledge of the revealed truths, has expressed its own belief many times over the course of the centuries, and since the bishops of the entire world are almost unanimously petitioning that the truth of the bodily Assumption of the Blessed Virgin Mary into heaven should be defined as a dogma of divine and Catholic faith ... we believe that the moment appointed in the plan of divine providence for the solemn proclamation of this outstanding privilege of the Virgin Mary has already arrived.
For which reason, ... we pronounce, declare, and define it to be a divinely revealed dogma: that the Immaculate Mother of God, the ever Virgin Mary, having completed the course of her earthly life, was assumed body and soul into heavenly glory.
Hence if anyone, which God forbid, should dare willfully to deny or to call into doubt that which we have defined, let him know that he has fallen away completely from the divine and Catholic Faith.