The story
Attested from 1900. The phrase, meaning "intense interrogation by police," likely refers to the Third Degree of Master Mason in Freemasonry, the ceremony for which included an interrogation. Alternatively, it reflects the practice of interrogation under torture, where three degrees of torture were recognized, of increasing intensity. In other contexts, three degrees of interrogation were recognised, with torture being the third degree.
Etymology from Wiktionary, CC BY-SA 4.0.