After the sack of Baghdad, which broke what little remained of the Abbasid Caliphate's power, the caliph's center of power was shifted to Cairo. But the Mamluks had no interest in sharing power with the Abbasids; they were little more than ornamentation for the sultan's court. And this remained the case right up until Selim's initial campaign into Syria turned into a full-blown conquest of the Mamluks (no one in the Ottoman court seemed able to predict how much of a pushover the Mamluks would prove to be; Selim's goal for the campaign was initially just Aleppo).

When Murad I proclaimed himself caliph, he wasn't really claiming to be the rightful descendant of the Abbasids by blood or anything. Rather, he was claiming that the Ottoman Empire, which had recently conquered Edirne and was now entrenched in Europe, was the rightful leader of the Islamic world.

Murad's proclamation was pretty ill-timed, though. He would die in Kosovo and his successor, Bayezid I, would suffer a catastrophic defeat at the hands of Timur (and die in captivity). The deaths of Murad and Bayezid, and the ensuing civil war when Bayezid's sons all started trying to kill one another, set the Ottomans back by three or four decades.

As with many of the more interesting proclamations by the Ottoman Empire, it was a legal fiction that was eventually made true by the fact that they pretty much exterminated their rivals. After all, by the time the dust from Selim's campaign settled, they held the three holy cities (Mecca, Medina, Jerusalem) and were still on the forefront of Islam's frontier. It turns out it doesn't really matter whether or not you're right when your opponents aren't alive to contest your statements -- just like when Selim justified his campaign against the Mamluks (who were also orthodox Sunnis like the Ottomans) by claiming "he who aids the heretic is himself a heretic." The original heretic in this case would be Shah Ismail of Persia.