The immediate answer to your question is that it was because Europe was a Christian continent, and Rome was the seat of the Papacy. The Popes were the heirs to the Throne of St. Peter, the first Pope and the disciple that Jesus entrusted his church to. This bestowed the Popes with significant powers and influence in Christendom, of which most of Europe was a part.

Now that we know why Rome was important, we can turn to why the monarchs were so interested in Rome, and there are several reasons. The first is that the Pope was the representative of God on Earth, and the supreme figure in Christianity. This is the most important factor, and the one I’ll focus on. The second is because the Pope had access to all the Papal wealth and resources, giving him non-spiritual influence too. The third reason was that the Pope was able to use all of this influence to become a significant player in European politics.

As representative of God, the Pope was able to take multiple actions which could hugely benefit the recipient of his attention. He could issue Papal Bulls, which could take precedence over the laws of states. One such Papal Bull, named the Inter caetera, was issued in 1493 by Pope Alexander VI, who had previously been known as Rodrigo Lanzol y de Borja, commonly translated as Rodrigo Borgia. You may have heard of him. The Inter caetera granted all the lands discovered beyond a certain point in the Atlantic to the Spanish Crowns of Castille and Aragon. Those crowns were held by Queen Isabella of Castille and King Ferdinand II of Aragon. They funded an expedition into the West by a Genoese man known as Christopher Columbus. So now, knowing what Columbus discovered, and what the Pope had just granted Isabella and Ferdinand, we can immediately see the value of the Pope as an ally. Especially, when the lands were granted, “by the authority of Almighty God […] to you, your heirs and successors”, which put the Spanish Crown in a position to become the most powerful kingdom in Europe for the next few centuries.

The Pope also had other powers, which were able to be exercised in a less beneficial manner. He could excommunicate people, and groups of people, regardless of status or size. This made for a useful threat in Christian Europe. His position as leader of the Roman Catholic church also meant that when a Pope endorsed a monarch, that King or Queen was granted the divine-right to rule. This essentially meant that that person was now above all judgement by any man, excepting the Pope. No one other than the Pope could claim to have the ear of God. This made monarchs nigh on infallible and protected their status. But the Pope could also remove this endorsement. A notable example of the power of religious titles is that of Henry VIII, who was granted the title “Defender of the Faith” by the Pope, then had it revoked after he split from the Catholic Church and was excommunicated. Once the new Anglican church was set up however, the title was reinstated by the English Parliament, and is retains by the British monarch to the present day. We can see from this how important divine-right was – even after the split from the Catholic Church, it was important that Henry retain religious links and superiority.

As the sovereign rulers of the Papal States, the Popes also had access to the wealth and resources of the Papacy, and the Papal Army. They were also the sovereign rulers of the Papal States, making them monarchs in their own right. The Church also owned land all across Europe – churches, monasteries, abbeys and the housing of important clerical figures such as bishops. These provided them with a certain degree of economic and regional influence. These were always useful in times of war or peace.

The Holy Roman Empire was ruled by the Holy Roman Emperor, who required the current Pope to approve him and coronate him. This guy was(supposedly) the supreme Catholic monarch in Europe, and by virtue of the title, heir to Charlemagne and ruler of the Germanic states. The Pope’s blessing being required was just one more reason Rome had such importance. Two generations after Isabella and Ferdinand were granted the possession of all those lands and territories Columbus discovered, their grandson became the Holy Roman Emperor Charles V.

So after all that, we see that Rome was important because it was the seat of the Papacy, who wielded more influence than any other person in Europe due to their office of Vicar of Christ and heir to the Throne of St. Peter. This in turn meant that the Pope could put monarchs into favour, giving them God given rights to rule, or remove them from favour, leaving them open to questions of their legitimacy. As well as that, the Pope had the wealth of the Church, and significant land holdings across Europe. He was also the man responsible for the coronation of the Holy Roman Emperor, who in turn was the leading Catholic monarch and the ruler of Germany. Lastly, as in the case of Isabella and Ferdinand, he was an invaluable patron, able to give away vast rewards with a single decree.

 

Resources:

Wikipedia gives a good basic overview for most of this. However, for more in-depth reading, there are these:

General Histories

Thomas Bokenkotter. A Concise History of the Catholic Church.

Bernard Hamilton. The Christian World of the Middle Ages.

Papal Power

Uta Blumenthal. The Investiture Controversy: Church and Monarchy from the Ninth to the Twelfth Century.

Constance Hoffman Berman. Medieval Religion: New Approaches.

John Thomas. The Western Church in the Middle Ages.

Conquest of the New World(I only hinted at it, but it is beyond fascinating):

Luis Rivera. A Violent Evangelism: The Political and Religious Conquest of the New World.

Stephen Greenblatt. Marvellous Possessions: The Wonder of the New World

 

Edited for formatting.