He had nerve damage on his hands, and thus had lost the fine motor control needed to perform brain surgery. That does not mean he could not use his hands for daily things. He did email people, for example.
In a deleted scene, not long before meeting the Ancient One, he treated this dog's broken paw:
During training, we see him using this hands quite nimbly - for example, when stealing books from the library while Wong listens to All the Single Ladies.
He spars with Mordo at one point. Midway through the movie he manages to operate the Eye of Agamotto, whose user interface is a bastard crossing of a Nintendo Power Glove and a safe dial.
As for the healing method proper, the Ancient One's introductory speech to Strange was all about it - there are many methods of healing other than modern western medicine. She shows him drawings of chakras as detailed in yoga and other Hindi practices, and if I recall well also a diagram of acupuncture points. Strange probably learned and was subjected to a combination of such practices as those and reiki, qi gong, etc., plus whatever alternative medicine practices exist only in the Marvel universes.
P.s.: more as a curiosity... In the comics, Strange has been in situations in which his magical powers were negated. In The Oath, Strange and one of his foes use a spell to completely negate all their magical powers, and then they have a fistfight. The fight makes it clear that, no matter how damaged Strange's hands are - if they are still damaged - he can still use them competently, even without magic. This has not been shown in the MCU, but the movies would probably follow along this should Strange ever be depicted in a situation where he cannot use any magic.