...how people [born into Judaism or Islam] get to the point where they accept the Word as true even as they're rejecting some of the things they've been taught in their birth-religion.This is a difficult and important question. Here are two passages that provide the beginnings of an answer. Spiritual Experiences 5709, quoted in Perspective on the “Two Foundations of Truth” talks about people who are “no longer able to be convinced from the Word” and indicates that, so far as these people are in good, truth from science may be able to open their intellects.
Similarly, Apocalypse Revealed 936, explains that the leaves of the tree of life in the Holy City that “were for the healing of the nations” mean rational truths
...by which they who are in evils and thence in falsities are led to think soundly, and to live becomingly.... [because] they who are in evils and thence in falsities cannot be healed by the Word, because they do not read it; but if they have sound judgment, they can be healed by rational truths.