。
With a little snap, I bit the thread clean through with my canine tooth.
Literal
Snap, [adverb-と] thread [obj-を] canine-tooth [with-で] bit-cut.
プチッ is a sharp, quick mimetic for a small snap or pop — pitch-perfect for the tiny sound of breaking a thread. The と after the mimetic (with a comma as stylistic pause) is the adverbial marker. 噛み切る is a compound verb 噛む + 切る 'to bite and cut through'; the stem+切る pattern productively forms 'to fully / completely X' readings (読み切る read through, 使い切る use up, 食べ切る eat up). The instrumental で marks 犬歯 'canine tooth' as the tool.