Hello! You asked why Danforth acts the way he does when evidence is presented against witchcraft in court.
In 17th Century Puritan New England, belief in the supernatural, especially belief in the Devil, was wide-spread. Life was hard in the colonies during this period; colonists were still reeling from wars between the English and the French on American soil, the threat of conflict between colonists and Native Americans was all too real, and a recent small-pox epidemic rendered Puritan spirits in a state of despair, worry, suspicion, and anxiety. Those unlucky enough to be accused of witchery were usually persons considered to be social outcasts by the larger Puritan community. The first condemned women were thought to be those who had indulged in extra-marital affairs and those who had birthed illegitimate children. The Puritan community believed it was their duty to root out what they considered unsavory elements from their society.
Danforth is the Deputy Governor. It is his duty to preserve the integrity of the Puritan court system. Any secular judgment meted out by the court would be seen by the larger Puritan community as comparable to divine judgment. Since the court was widely viewed as a vehicle for rooting out witchery and devilish elements from Puritan society, the sanctity and reputation of the court was fiercely defended by officials like Danforth. As a community leader and an accessory to divine justice, he must give no evidence that his judgment was not to be trusted. Furthermore, the court was also used as a vehicle for keeping the populace cowed under the authority of its leaders; the court helped its leaders protect their own power structures within the community.
Danforth: Peace, Judge Hathorne. Do you know who I am, Mr. Nurse?
Francis: I surely do, sir, and I think you must be a wise judge to be what you are.
Danforth: And do you know that near to four hundred are in the jails from Marblehead to Lynn, and upon my signature?
Francis: I -
Danforth: And seventy-two condemned to hang by that signature?
Francis: Excellency, I never thought to say it to such a weighty judge, but you are deceived.
In short, Danforth doesn't want the community to see him as fallible in any way. When Proctor presents him with the deposition containing the signatures of ninety one land-holding farmers testifying to the good character of his wife, Elizabeth, and two other women, Danforth is not overly impressed. Although he doesn't overtly speak in support of Parris' argument that the deposition is an attack against the court, he nevertheless reserves the right to invade the privacy of the signers, claiming that they should have nothing to fear from being called in for questioning if they were innocent, 'covenanted Christians.'
In summary, Danforth is only interested in defending his reputation and his power; he does so under the guise of being an arbiter of justice. Furthermore, Danforth is not too interested in going down in history as the guy who got bamboozled by a group of women and a few teenage girls.
Thanks for the question.