Crucibles are used to heat materials to extremely high temperatures. The connotations of this type of device include notions of pressure and a reduction to "the elemental".
In Miller's play, the townspeople are pressured to comply with the demands of the court and, in doing so, many of them demonstrate their true natures. Even the outsider, Hale, is reduced by the "crucible" of the situation to a state wherein his true nature is clearly shown. He is not nearly as certain or as righteous in his heart as he initially acts in public.
The greatest example of a character reduced to his fundamental nature is John Proctor.
Proctor is challenged in this play to act in ways that are more forthright than he may have acted under more normal circumstances. He is pressed to decide who he is, at bottom, on several occasions. In order to save his wife, he decided to sacrifice his honor and admits to his affair with Abigail. In his decision to go to the gallows instead of signing a false confession, he is again tested to prove who he is in a fundamental way.
He is at his most self-aware in his final speech when he realizes the importance of maintaining his integrity.
The idea of the crucible prepares us to recognize the revelations of character as "elemental" truths that have been produced or revealed by the process of adding heat in a highly confined space.