The mechanism that destroys Dasharatha is set years earlier and forgotten. In a battle against the asuras, Kaikeyi accompanied him as his charioteer and saved his life, and he granted her two boons to be named whenever she chose. She named neither, and the debt sat unpaid.
On the eve of Rama's installation as heir, her attendant Manthara persuades her that the elevation of Kausalya's son will reduce her and her own son to servitude. Kaikeyi goes to the anger chamber, and when the king comes she asks for the two boons: that Bharata be installed instead, and that Rama be exiled to the forest for fourteen years.
Dasharatha's reaction occupies a long stretch of the poem and it is all refusal without any means of refusing. He argues, he offers her anything else, he pleads, he collapses, he threatens to disown her, and he cannot simply say no, because a king's given word is the ground on which the entire order of obligation in the epic stands. If his promise can be broken because keeping it is unbearable, nothing else in the poem holds.
Rama, told of it, agrees immediately and without argument, which is how the Ramayana establishes him. He does not need to be persuaded, and his father's ruin is complete the moment his son shows how easy the obedience is.
The king dies within days of the departure, calling for Rama. The exile has fourteen years to run and the man whose word imposed it does not survive the first week of it.