How did the Catholic Pope manage to become more powerful than Kings in old Europe?

This question needs a great deal more substantiation, but we can look to three fundamental factors that provided the base for the power of the Church and the Papacy in Medieval Western Europe:

• Catholic Justified Submission - precursor to The Divine Right of Kings was perhaps the most important single factor empowering the Church and the Pope in Medieval Europe:

Originating in Europe, the divine-right theory can be traced to the medieval conception of God’s award of temporal power to the political ruler, paralleling the award of spiritual power to the church. By the 16th and 17th centuries, however, the new national monarchs were asserting their authority in matters of both church and state. King James I of England (reigned 1603–25) was the foremost exponent of the divine right of kings, but the doctrine virtually disappeared from English politics after the Glorious Revolution (1688–89). In the late 17th and the 18th centuries, kings such as Louis XIV (1643–1715) of France continued to profit from the divine-right theory, even though many of them no longer had any truly religious belief in it.

• Divine right of kings - Catholic Justified Submission

Catholic thought justified submission to the monarchy by reference to the following:

The Old Testament, in which a line of kings was created by God through the prophecy of Jacob/Israel, who created his son Judah to be king and retain the sceptre until the coming of the Messiah, alongside the line of priests created in his other son, Levi. Later, a line of Judges (who were not kings as they only had the power to provide insight to the people and not to take action to enforce their rulings) was created alongside the line of High Priests created by Moses through Aaron. Later still, the Prophet Samuel re-instituted the line of kings in Saul, under the inspiration of God.

The New Testament, in which the first pope, St. Peter, commands that all Christians shall honour the Roman Emperor (1 Peter 2:13-17), even though, at that time, he was still a pagan emperor. Likewise, Jesus Christ proclaims in the Gospel of Matthew that one should "Render unto Caesar the things which are Caesar's"; that is at first, literally, the payment of taxes as binding those who use the imperial currency.

The endors*m*nt by the popes and the church of the line of emperors beginning with the Emperors Constantine and Theodosius, later the Eastern Roman emperors, and finally the Western Roman emperor, Charlemagne and his successors, the Catholic Holy Roman Emperors.

Since during the period in question, religion in (Western) Europe was the virtual monopoly of the Catholic Church headed by the Pope, the sanction of the Pope was in many respects the source of any monarch's claim to the throne, and the power of any monarch was a function of that monarch's relationship with the Church.

Perhaps the most well known historical example of the intimate bond between Papal power and Monarchic Power is the English Henry VIII's formation of the Anglican Church in concert with Thomas Cromwell, the outcome of what came to be known as the King’s great matter

From 1527 Henry pursued what became known as “the King’s great matter”: his divorce from Catherine. He convinced himself that his first marriage had been against the divine law; that is, against the biblical injunction (Lev.) forbidding marriage with a brother’s widow.

He appealed to Rome for a declaration of annulment. Popes had usually obliged kings in such matters, but Henry had picked both his time and his case badly. He was asking Pope Clement VII to help him discard the emperor’s aunt, but Clement, the emperor’s prisoner in 1527–28, never thereafter dared resist Charles, whose powerful feelings of familial honour and public prestige barred any concession to Henry’s wishes. Moreover, the pope’s reluctance was increased by the fact that he was being asked to declare illegal an earlier exercise of papal power—which had licensed Henry’s marriage to his brother’s widow—of a kind that brought a good deal of money to the papal coffers....

Action called for a revolution, and the revolution required a man who could conceive and execute it. That man was Thomas Cromwell, who, in April 1532, won control of the council and thereafter remained in command for some eight years. The revolution consisted of the decision that the English church should separate from Rome, becoming effectively a spiritual department of state under the rule of the king as God’s deputy on earth. The revolution that he had not intended gave the king his wish: in January 1533 he married Anne Boleyn; in May a new archbishop, Thomas Cranmer, presided over the formality of a trial that declared the first marriage annulled; in September the princess Elizabeth was born. The pope retaliated with a sentence of excommunication; it troubled no one.

As long as Henry was beholden to the Pope in Rome, his situation as King was compromised. The only solution was to break ties with Rome and create a new church, which recognized Henry as King by Divine Right.

• The Power of the Purse: Although indeed the Pope and the Church were "not exactly military men", the Church controlled massive amounts of wealth in Medieval Europe, accumulated through tribute, tithes, indulgences, and bequeathals and gifts that accumulated to the extent that the Church was the largest landowner in Europe and controlled vast stores of treasure, sequestered away in the Rome, and in churches, monasteries, convents, etc throughout Europe. (Some legends and traditions also claim that treasures looted by the legions of Rome are hidden in the bowels of the Vatican store-rooms, but I don't know how much of that is historically verifiable.)

The situation in England typified what was common in Europe at large during the period in question:

Typically, 11th and 12th century founders had endowed monastic houses with both 'temporal' income in the form of revenues from landed estates, and 'spiritual' income in the form of tithes appropriated from parish churches under the founder's patronage. In consequence of which, religious houses in the 16th century controlled appointment to about two-fifths of all parish benefices in England disposed of about half of all ecclesiastical income and owned around a quarter of the nation's landed wealth.

Along these lines, with the founding of the Anglican Church Henry instituted the Dissolution of the Monasteries

A set of administrative and legal processes between 1536 and 1541 by which Henry VIII disbanded monasteries, priories, convents and friaries in England, Wales and Ireland, appropriated their income, disposed of their assets, and provided for their former members and functions. He was given the authority to do this in England and Wales by the Act of Supremacy, passed by Parliament in 1534, which made him Supreme Head of the Church in England, thus separating England from Papal authority, and by the First Suppression Act (1536) and the Second Suppression Act (1539).

Wealth and territory bequeath power; Kings need money.

• Rome was already the seat of power in Western Europe prior to the rise of the Church, as the capital city of the Roman Empire: Europe was accustomed to looking towards Rome for leadership and authority. In many respects, the Papacy and the Church filled the vacuum of power left in Western Europe after the fall of the Roman Empire. Although the Church did not have the legions of the empire, it had other ways of expressing power, as we have mentioned, thus paving the way for the Church to take up where the Roman Empire had left off, as the nexus of power in Western Europe.