The transmuter was, essentially, an amplifier, permitting mental impulses to shape the physical world. The explorers Sylvia and Korob used it to enter our galaxy from wherever they originated. It also enabled them to restructure matter, transport themselves from place to place, exert mind control, affect the physical world in various ways, and to perform other feats, including some that resembled sympathetic magic.
Korob's transmuter appeared in the form of a magical wand; this was consistent with the image he wished to project of himself as a sorcerer, an image pulled telepathically from the human subconscious. He never voluntarily released the device, it is possible that he could control it properly only when in physical contact; certainly, Sylvia could not control it when she was not in contact. The head of the instrument, a large faceted crystal, glowed when Korob used its power to restructure matter.
Sylvia possessed a less powerful instrument; hers took the form of a crystal pendant, and was the only thing consistent across all her forms. It was evidently far less powerful than Korob's device, as she was not able to use it to alter the fortress-like headquarters he had created. Sylvia also described Korob's device as "holding the key"; it may be that all the advanced functions required it. Whatever the reason, Sylvia was desperate to possess the transmuter after Korob died, using various combinations of trickery and threat to persuade Captain Kirk to surrender it. Ultimately, Kirk destroyed the device, which undid everything Sylvia and Korob had created; it is possible that Sylvia's device was a tap that permitted her limited access to Korob's transmuter.
The transmuter also possessed a power that strengthened them and was like a life support system to them. When it was destroyed, they died in less than a minute. Whether it was due to an inimical property of our galaxy or due to natural fragility in a different environment remains a mystery. (TOS: "Catspaw")