The situation that Elizabeth is faced with in act three is a very difficult one. Proctor has just accused Abigail of being a whore--a weighty charge, to be sure. Danforth wants more proof than Proctor's word. Abigail denies the affair, and so, based on John's own assertion that his wife fired Abby over the affair, Danforth brings Elizabeth in. John, confident in his wife's honesty, tells Danforth that she "has never told a lie" in her life. Based on Elizabeth's honesty, Danforth decides to either legitimatize Proctor's claim about Abby, or to dismiss it.
The reason that John accused Abby in the first place was in a desperate attempt to discredit her reputation. She was trying to pull stunts in the court again, to make Mary Warren look like a witch. Reason hadn't worked, and the judges were believing Abby. John needed to show that Abby was not to be trusted--she was not a holy person, working as an instrument in god's hands to bring witches to justice. She was, instead, a vain girl determined to get Elizabeth (who fired her) out of the way so that she could be with her love, John.
If John's claim that he had an affair with Abby is true, it discredits every single testimony that Abby gave against the supposed "witches." It redeems Elizabeth, because it proves that Abby was just trying to kill her to be with John. If John's claim about the affair is judged as false, then John will be seen as a liar who is trying to overthrow the courts--he will be seen as a huge threat to their authority and "justice." So, Elizabeth is brought in, and tells the only lie of her life, which is that the affair didn't happen. It is an "understandable lie to tell" (as Hale states), because she didn't want the town knowing her husband cheated on her. However, it is also a very unfortunate lie, as it empowers Abby to turn on Mary again, which ultimately leads to John himself being arrested. If Elizabeth had told the truth, she would have been freed, and Abby would have been the one to be brought under questioning from the courts instead.
I hope that those thoughts helped; good luck!