Why did Lord Rama ask mother Sita for Agni Pariksha?
In this chapter of the Yuddha Kanda of the Ramayana, Rama explains why he made Sita walk through fire, despite having complete trust in her:
Seetha certainly deserves this pure factory ordeal in the eyes of the people in as much as this blessed woman had resided for a long time indeed in the gynaecium of Ravana. The world would chatter against me, saying that Rama, the son of Dasaratha, was really foolish and that his mind was dominated by lust, if I accept Seetha without examining her with regard to her chastity.
I also know that Seetha, the daughter of Janaka, who ever revolves in my mind, is undivided in her affection to me. Ravana could not violate this wide-eyed woman, protected as she was by her own splendour, any more than an ocean would transgress its bounds.
In order to convince the three worlds, I, whose refugee is truth, ignored Seetha while she was entering the fire. The evil-minded Ravana was not able to lay his violent hands, even in thought, o the unobtainable Seetha, who was blazing like a flaming tongue of fire. This auspicious woman could not give way to the sovereignty, existing in the gynaecium of Ravana, in as much as Seetha is not different from me, even as sunlight is not different from the sun. Seetha, the daughter of Janaka, is completely pure in her character, in all the three worlds and can no longer be renounced by me, as a good name cannot be cast aside by a prudent man. The salutary advice of you all, the affectionate guardians of the world, who are saying what is conducive to our good, must be certainly carried out by me.
So it was done in order to demonstrate to the rest of the world that Rama wasn't taking in an unfaithful wife, even though Rama personally knew she was faithful.
Of course, this still wasn't enough to make the people of Ayodhya believe that Sita had been faithful, which is what ultimately led to Sita's exile.
By the way, on a side note according to the Skanda Purana, it is Vedavati, not the real Sita who goes into the fire, as I discuss in my answer here.
Note: “The question: Why did Lord Rama ask mother Sita for Agni Pariksha?” is licensed by Stack Exchange Inc (https://hinduism.stackexchange.com/); user contributions licensed under CC BY-SA.