Rashi: He wrapped it [so it would be easy to strike the water with it]. Radak - this is like "bi'Glomei Techeles" (Yechezkel 27:24), i.e. wrapped garments.

Malbim: He made it a Golem (physical substance that is prepared to be changed and made into a Kli as the artisan wants) - "Galmi Ra'u Einecha" (Tehilim 139:16).

Malbim: Chachamim say that Eliyahu's body was purified from earth and water, the two heavy elements of the four. Only fire and wind remained, so he was able to ascend. There was a chariot of fire, and he ascended in a wind, and his body remained above. When the angel told him to eat Ugas Retzafim (a cake of flames) and drink the flask of water, his body was purified from earth. Now he wanted to separate from water, which also descends below. When one does a miracle in his lower world (body), this increases his strength also in the great world. If he makes the nature of his body submissive to Divine matters, against its nature, also the general nature is drawn after him. Eliyahu wanted to nullify the element of water from his body. He used his scarf, for it is a metaphor for his physical garment [of his soul, i.e. his body].

Malbim: Nullification of nature in his small world obligates nullification of nature in his big world.

Radak (1): Hashem told Eliyahu that he will be taken there. Perhaps it is where Moshe died, for Eliyahu's level was close to Moshe's. He was shown honor on Har Sinai like Moshe, and fasted for 40 days, like Moshe.