In such situations the pope was not representing either party, both parties signed it separately themselves as the Wikipedia for the Treaty of Tordesillas notes. The Pope in such contexts was acting more as a arbiter of Europe. The Latin and Catholic Papacies (Catholic in historical terminology is the Roman church after the Council of Trent) had a prominent role in international peace treaties and crusading alliances in Europe. Because peace is sacred and the Pope is the head of the church (for those who recognise him) he is, of course, the perfect individual to mediate and especially to sanctify peace treaties, giving them particular weight which reassures both parties that the other is more likely to keep it. The Italian Renaissance, occurring when the Treaty of Tordesillas was signed, was a period of reassertion of papal authority where popes were both rebuilding Rome and the Vatican finances and rebuilding their international status, after both had declined due to conflict and scandal. Thus, as well as providing more assurance of the treaty and being in keeping with their role, it was specifically what a pope at the time wanted to be seen doing; something important and respectable.

One of the bases of the pope's role was in traditional virtues, mainly the Platonic or Cardinal Virtues: fortitude, i.e patience and perseverance, justice, i.e moral vigor, temperance, i.e restraint and purity,, and prudence, i.e. logic and sensible judgement, ("The Ideal Renaissance Pope: funeral oratory from the papal court", JM McManamon 1976). You can see how these values relate to peace. Quote from McManamon; "virtue is especially related to active works of charity, while an eloquence to persuade his fellow men to peace and concord and a moral life is the hallmark of his wisdom." (bolding my own). McManamon describes peace in Christian Europe as one of the five central goals of the Renaissance papal revival. Peace in particular is one of the more moralistic and international goals, which makes it arguably more important compared to more narrow (though significant) goals like patronising great art and rebuilding Rome. While the Papal States had an army which was deployed at various times and the pope could lead holy leagues international diplomacy was something of a speciality for the papacy since it was not a great military power. Not only at the conventional negotiating table, but also through influence on international clerical appointments, church resources, dogmas and policies, and the issuance of documents like indulgences, dispensations, excommunications and papal bulls. At a time when religiosity was taken so seriously and literally, religious power was a significant leverage, especially when attached to money and land.

Pope Clement VIII is a good later example, being instrumental in ending the French Wars of Religion by accepting Henry IV into the Catholic Church, and conversely served as a key broker for the formation of the European Christian alliance against the Ottomans in the Long War (1591-1606). While Henry VI still had to fight, and was eventually assassinated by a Catholic fanatic, and while the Hapsburgs had long been in conflict with the Ottomans, his intervention on both issues carried major significance. The alliance between the Spanish Hapsburgs and the House of Guise was always tenuous given they both had their own ideas about prefered royal candidates, so taking the religious justification out of the war helped to weaken the already shaky Catholic League, which was defeated soon afterwards. Similarly, in the case of the Long War, it was Pope Clement who encouraged the Hapsburgs and the Princes of Transylvania, Wallachia and Moldavia to formally coordinate together, although he failed to gain the formal cooperation of Poland he desired. Clement sent some of his own money and soldiers to the fight in Hungary during the Long War.

On the specific issue of Tordesillas it is worth noting it was a moral issue twice only, not only as an agreement to avoid conflict, but also an agreement about the moral status of colonialism and slavery. While it did not mention slavery per se, it did give both crowns absolute rights over the Americas and the people there, and popes had approved slavery explicitly before, such as the then-recent 1454 grant by Pope Nicholas to King Alfonso V of Portugal to practise slavery in his new territories in North Africa, quote: " to invade, search out, capture, vanquish, and subdue all Saracens, and other enemies of Christ... and to reduce their persons to perpetual slavery". So it was important to get that kind of papal approval since both the Spanish and Portuguese wanted to invade, search out, capture, vanquish, subdue and indeed, reduce people to slavery.

Hope this helps.