The “emperor” in question wasn’t actually the last Byzantine Emperor - today we generally consider the last emperor to have been Constantine XI, who died in the battle for the city in 1453 and had no children. He did have a brother though, Theodore Palaiologos, who was the ruler of the other remaining Byzantine territory, the Despitate of Morea (roughly the Peloponnesus in Greece). The Morea actually held out against the Ottomans for several more years until 1460. Theodore died in 1465 and his claim to the Morea passed to his son Andreas Palaiologos.

Andreas, like many other refugees from the former Byzantine Empire, fled to Corfu and then to Italy, and he ended up living in Rome, supported by a pension from the Pope. His brother Manuel and his sister Zoe came with him. His other sister had already married the Despot of Serbia but then Serbia was conquered by the Ottomans as well. Zoe eventually married Ivan III, the tsar of Russia, and Manuel actually ended up in the service of the Ottoman emperor Mehmed in Constantinople.

Andreas however remained in Rome and began using the title of Emperor as well as continuing to claim the Despotate of Morea. For the next few decades he tried to organize a military expedition against the Ottomans but no one was willing to join an obviously hopeless cause. In 1475 he tried to sell his titles to the king of Naples, the duke of Burgundy, and maybe the duke of Milan; the duke of Milan was assassinated in 1476 and the pope’s expenses in the resulting wars that broke out in northern Italy meant that he could no longer afford to pay Andreas’ pension.

In 1480 he visited his sister Zoe in Moscow but she had no money to give him. He also visited Naples, which would have been the logical place to launch an invasion of Greece. Venice had also been fighting the Ottomans in the Aegean and Andreas thought he could count on their support as well. Mehmed the Conqueror died that year so it might have been a good time to invade, if they had been able to organize anything. But King Ferdinand of Naples wasn’t able to help, and Venice’s war with the Ottomans had ended in a truce so the Venetians weren’t interested in breaking it.

In 1490 he visited Russia again, but found no help there either. He visited King Charles VIII in France in 1491 and King Henry VII in England in 1492. Only Charles was willing to help although it’s possible that the negotiations were conducted without Charles’ direct involvement. Andreas and the French agreed that Charles would attack the Ottomans and restore Andreas as the Despot of Morea. Andreas would give up his claim to the imperial throne if Charles could conquer Constantinople. Unfortunately for Andreas, Charles was far more concerned with invading Italy and conquering Naples, which he did from 1494 until his death in 1498. Charles’ successors actually continued to claim the title of Byzantine emperor for a little while, but as far as Andreas was concerned, the transaction never took place because Charles never conquered Constantinople.

Andreas remained in Rome where he died in poverty in 1502. In his will, he bequeathed the imperial title to Ferdinand and Isabella, but he never visited Spain and they had no knowledge of this until after Andreas died. Since they didn’t seek the title, and it was sort of legally questionable whether they had any claim to it, neither they nor their successors ever officially used the title.

So why them? Well we don't really know, but presumably it was because he had already been rejected by everyone else, and he knew that at the end of his life he might as well just give the title away since no one else had been willing to buy it from him.

Sources:

Jonathan Harris, "A Worthless Prince? Andreas Palaeologus in Rome, 1465–1502", in Orientalia Christiana Periodica 61 (1995), pp. 537–554

Donald M. Nicol, The Immortal Emperor: The Life and Legend of Constantine Palaiologos, Last Emperor of the Romans (Cambridge University Press, 1992)

Kenneth M. Setton, The Papacy and the Levant (1204–1571), Volume II: The Fifteenth Century (American Philosophical Society, 1978)

I haven't been able to find this one, but Andreas' will is apparently discussed (in German) in P.K. Enepekides, "Das Wiener Testament des Andreas Palaiologos vom 7. April 1502", in Akten des 11. Internat. Byzantinisten-Kongresses 1958 (C.H. Beck, 1960), pp. 138–143