In this case the ability of Illusionist's Bracers will trigger and when it resolves, Dominating Licid's ability is copied. When the copy resolves, Dominating Licid loses the activated ability (but this doesn't prevent the other ability still on the stack from resolving [C.R. 112.7a]), becomes an Aura, and is attached to the targeted creature. But since Dominating Licid isn't a creature anymore (C.R. 205.1a), and because an Equipment "can't legally be attached to an object that isn't a creature" (C.R. 301.5), Illusionist's Bracers will fall off as a state-based action (C.R. 704.5n). When the original resolves, Dominating Licid again loses the ability, again becomes an Aura, and is attached to the creature targeted by that ability (which may be different from the copy's target). Note that a player can't gain control of two creatures this way (if a different creature was targeted for the copy than for the original), since Dominating Licid will attach to one creature, then another, and the effect setting the creature's controller applies only to the creature it enchants (C.R. 611.3a). Moreover, to completely end the effects of Dominating Licid's abilities, UU must be paid, since the two effects overlap.
EDIT (Sep. 28): One rule was renumbered with Ixalan.
EDIT (Sep. 28): One rule was renumbered with Ixalan.