How does the title of Arthur Miller's The Crucible highlight key ideas explored in the play?

The word crucible has several definitions that fit the ideas explored in the play. First, a crucible can be a small cup or bowl in which some substance can be melted using high temperatures; it can also refer to a severe test; finally, it can refer to a situation in which volatile forces can come together to provoke change.
To begin, then, we can read the title symbolically, where Salem becomes like a crucible, in which relationships of the town are "heated up" and tested by fear and hysteria that result from the girls' witchy activities in the forest. At the beginning of the play, Reverend Parris is especially upset with his niece Abigail, telling her "if you trafficked with spirits in the forest I must know it now, for surely my enemies will, and they will ruin me with it." He believes that a faction who wishes to "drive [him] from [his] pulpit" is working against him and will use this new information to do just that. In his own fear, he invites Reverend Hale, a known witch-hunter, to town, increasing people's anxiety and fear of danger. Hale's appearance results in Tituba's accusation of two local women, which leads to Abigail and the other girls accusing yet more people.
Certainly, these events seem to constitute a test. Will people retain their wisdom or will they give in to their fear? Will people remain loyal and honest, or will they turn on friends and neighbors? John Proctor's frustration at the end of act 2 provides evidence for the latter. He asks,

Why do you never wonder if Parris be innocent, or Abigail Is the accuser always holy now? Were they born this morning as clean as God's fingers? I'll tell you what's walking Salem—vengeance is walking Salem. We are what we always were in Salem, but now the little crazy children are jangling the keys of the kingdom, and common vengeance writes the law!

The people of Salem are being tested, and they are failing. They are allowing themselves to be completely taken by the children's accusations, many doing so without questioning or critically thinking. Some are also exploiting the children for their own selfish, greedy reasons, often to do with the stealing/buying of land. Fear and greed seem to compel them to turn on people they've known their whole lives. They are, indeed, failing this test.
Certainly, volatile forces combine here to provoke change. In the end, most people seem to realize that they have responded to fear, hysteria, and greed by failing to exercise good sense and compassion. They begin to ask for it; however, Deputy Governor Danforth will not listen. Reverend Hale demands the pardon of the convicted. Reverend Parris begs for a delay, at least. Parris says,

it were another sort that hanged till now [....]. Let Rebecca stand upon the gibbet and send up some righteous prayer, and I fear she'll wake a vengeance on you.

Respect for Danforth's authority has all but disappeared, and people now feel more compassion for the convicted—now seen as victims—rather than the magistrates, who are now viewed as antagonists.
https://www.merriam-webster.com/dictionary/crucible