What does this quote mean and who said it? "I come to do the Devil's work. I come to counsel Christians they should believe themselves."Who is...

The speaker of the quote is Rev. Hale in Act IV, Scene I.  You do have a typo in the quote though.  It should read, "I come to do the Devil's work.  I come to counsel Christians they should belie themselves."  Hale is speaking to Danforth telling him that he has returned to convince those Christians who stand accused to lie and save their lives.  The circumstance behind the quote is that Rev. Hale sees the truth behind the girls' pretense in crying witchcraft on innocent people.

This quote is important in revealing Rev. Hale's character as coming full circle from the beginning of the play.  When he first entered Salem, he came with the "weighty authority of his books."  Now, he has returned to reverse the damage caused by his knowledge of witchcraft.  Hale has returned with the same energy that he had when he first entered Salem Village.  Even though he is a Christian, he feels just in counseling other Christians to lie to an unjust court system.