There are at least two ways that the tragedy and action of the play, The Crucible, are related to an idea of catharsis.
The story of John Proctor, central to the larger story of the play, is probably the most compelling example here. From the opening scene of the play, Proctor's affair with Abigail is hinted at and is presented as a poorly kept secret. Parris suspects that Abigail was "let go" at the Proctor house due to something dishonorable and this suspicion clings to Abigail and festers in the town.
Proctor is given the opportunity to confess and to bring this painful secret to light. He oscillates as to whether or not a confession would be a catharsis, freeing he and his wife from the pain of the secret or if a confession would do more harm to their damaged marriage.
Regardless of the ultimate confession and its results, we can clearly see Proctor's story as being related to catharsis.
A second way to relate this story to catharsis is by looking at all the petty feuds that spring up through the guise of the witchcraft scandal. In a community defined by restraint, there is quite a bit of material under the surface and the scandal brings much of it out in a communal catharsis.