The oath being said
An Eternal Girl is a Selector who becomes their ideal self, the person capable of fulfilling their wish. Becoming one is the goal of a Selector.
A Selector becomes an Eternal Girl when certain conditions are fulfilled. The conditions for an Eternal Girl always require that the Selector win a number of battles; the number varies from wish to wish. There may be other conditions required for becoming an Eternal Girl, aside from the number of battles. When a Selector fulfills these conditions and is about to win a Selector battle, the LRIG has the Selector recite an oath: "I accept the Laws of the Selector. And now, a new Eternal Girl will be born!" The LRIG and selector finish off the opponent.
However, when this occurs, the Selector does not keep their own body: instead, their LRIG inherits their body, while the Selector becomes a new LRIG. In this way, the Selector becomes the person capable of fulfilling their wish, but in a way unintended by the Selector. The LRIG who inhabits the Selector's body is still obligated to fulfill the Selector's wish, and cannot simply abandon the wish after becoming human.
As demonstrated in Episode 12, the oath is required to be repeated in its entirety for the wish to be granted. It is possible for a Selector's wish to interrupt another Selector from becoming an Eternal Girl.