What does Prospero do to bring the mariners to the island in The Tempest?

Prospero, a powerful magician, uses his command over the spirit Ariel to cause the titular tempest that shipwrecks the mariners on the shores of the island on which the play is set.
The play opens with the sailors fighting the storm in act I, scene i. It is not until scene ii that the audience meets Prospero and Ariel and the tempest is revealed to have been supernatural in origin through the following exchange:

PROSPERO
Hast thou, spirit,Perform'd to point the tempest that I bade thee?
ARIEL
To every article.I boarded the king's ship; now on the beak,Now in the waist, the deck, in every cabin,I flamed amazement: sometime I'ld divide,And burn in many places; on the topmast,The yards and bowsprit, would I flame distinctly,Then meet and join. Jove's lightnings, the precursorsO' the dreadful thunder-claps, more momentaryAnd sight-outrunning were not; the fire and cracksOf sulphurous roaring the most mighty NeptuneSeem to besiege and make his bold waves tremble,Yea, his dread trident shake.

Prospero's goal in sending Ariel to cause the storm was always to draw the sailors to the island, never to actually drown them or do them any physical harm. So after wreaking the havoc that Ariel describes in detail above, the spirit then ensured that the shipwrecked men made it safely to the shores of Prospero's island so that he could begin to exact his revenge.

Prospero used magic to bring the mariners to the island.
To explain that in a bit more detail, Shakespeare's play begins with a ship facing a storm, the "tempest" that gives the play its title. The storm is so intense that the sailors are afraid it will sink or destroy the ship. In fact, there's a tremendous sound in Act I, Scene 1, which makes the sailors think they really are about to die.
This was all orchestrated by Prospero, who sent Ariel, a spirit in his service, to shape the storm. Ariel made the wind, lightning, and thunder. Ariel also made sure everyone made it to the island safely, so you could say the combination of Prospero's magic, his desire for revenge, and his sense of responsibility brought the sailors to the island.