While the Knights Hospitaller set out to "take care of the poor and sick" at their inception, by the end of the Crusades they had become a genuine military order akin to the Knights Templar. When they were driven out of the Holy Land, they took up residence in Rhodes not to guard hospitals, but to continue crusading.

However, they were matched up against the emerging power of the Ottoman Empire, and there was only so much they could do. The most effective strategy the Knights could employ was one of maritime raiding. In command of a small fleet of galleys, they began to attack ships in the Eastern Mediterranean. While they focused on Ottoman and other Muslim ships, Christian powers weren't exempt, with the Venetians considering them to be pirates instead of holy warriors.

By the late 1400s and early 1500s, the Knights were becoming increasingly disruptive. Attacks on grain transports had caused food prices to skyrocket in Istanbul, and the Knights' habit of enslaving pilgrims bound for Mecca was greatly embarrassing to the Ottomans. The fact that Rhodes was so close to Ottoman territory was particularly galling, so the Ottomans took Rhodes in 1522, 42 years after they had first attempted its capture.

As for why they attacked Malta: the Knights had begun to attack Ottoman shipping again after several decades of relative calm. While the Knights' new fleet was tiny and posed little threat, the Ottomans were at the height of their power and wanted to both destroy the Knights and demonstrate their ability to project their power far from their borders.

TL;DR: The Knights of Rhodes were a bunch of pirates, not medics, so the Ottomans attacked them to protect their economy and their pride.

Source: Roger Crowley, Empires at Sea: The Siege of Malta, the Battle of Lepanto, and the Contest for the Center of the World (New York: Random House, 2008)

The Knights were effectively pirating Ottoman shipping, particularly between Egypt and Istanbul. Rhodes is basically right on some of the major shipping lanes in the East Mediterranean and to let a hostile state sit as a thorn in the side of the Ottomans was out of the question.