Portuguese legal position in the dispute was built upon Treaty of Alcáçovas from 1479, confirmed by pope in his bull Aeterni regis, and also papal bulls from 1450s Romanus Pontifex and Dum Diversas granting them rights to newly discovered lands, albeit not clearly defined

First a little background, this treaty of Alcáçovas was the peace conclusion of War of Castillian succession which rival claims to throne of Castille, featuring Isabella aided by her husband Ferdinand of Aragon on one side, and Portuguese king Afonso V through a marriage to his niece Joanna la Beltraneja(Juana de Trastámara), on the other.

The war was pretty much decided on land, in Battle of Toro, and resulted in Isabella getting the throne of Castille and Portuguese renouncing all claims.

However the conflict had another, naval, dimension.

Portuguese had since as early as the 1415, started sailing (and sometimes establishing forts) further and further south down the African cost. In 1440s they already reach Sub-Saharan Africa in Senegal and Gambia and found lucrative trade in slaves, gold dust, ivory etc. In 1450s they obtained papal bulls granting them exclusive rights (and control) to trade (and slavery).
In 1470s in another wave of discoveries they reached Guinea and found even more lucrative trade opportunities there. Isabella wanted a piece of this trade and during the war, she sent a fleet to trade there (in spite of Portuguese legal rights). Portuguese found out, and sent their own fleet, which intercepted the Spanish fleet there and captured it, thus securing naval control of Africa. At the same time Portuguese coveted and schemed to take control of Canaries from Spain, but ultimately failed in any military endeavor.

So Treaty of Alcáçovas, with the usual settling of claims to the throne, was used to determine the possessions in the Atlantic. Spain retained Canaries, and Portugal confirmed all it's possessions (Madeira, Azores, Africa) and exclusive right to discover and claim all lands south of the Canaries. This specific wording of "south" was the key of why the Portuguese claim to America was unsuccessful, as the lands Columbus discovered were not "south" of Canaries (that and pope Alexandar VI being Aragonese, and a little sympathetic to Spanish, confirming all their claims in his bull Inter caetera)

Also quite important was that at the time 1492-1494, there was still the mix up of America being actually Asia and India. And the Portuguese while maybe not caring about islands to the West were worried that Spansih would be granted rights to Asia and India, which was something very important to the Portuguese.

---

So to wrap up, the Portuguese felt entitled to the New discovered land which was only later discovered to be Americas and not India because:

• they had legal documents giving them what they thought was control of all new lands, but had a technicality of specifying only lands south of Canaries
• they at first thought (or wanted) that the newly discovered land was south of Canaries, thus theirs even legally
• they were the ones running the show of discoveries for almost 80 years, and felt one it was their just right. Columbus himself was former employee of Portuguese, and where he learned the ocean sailing skills
• they had the naval supremacy (or at least thought they had) to get away with it

Ultimately, with the Spanish being basically in the right and having current pope sympathetic to them, but with the Portuguese having an upper hand in naval strength and self-proclaimed moral right to this land, the parties agreed to diplomatically settle the dispute. The Treaty of Tordesillas was beneficial to both parties, as Spanish confirmed rights to America, while Portuguese pushed the border west (later giving them Brazil), secured India as still open to whoever comes first, and getting more then the papal bull Inter caetera was giving them