How would this happen? In what order were things lost? Reading these verses carefully we learn what was lost first. The gospel—the basic principles of the gospel were lost first: "they have taken away from the gospel of the Lamb many parts which are plain and most precious" (1 Nephi 13:26). And once a basic knowledge of the Atonement and repentance was lost, what else was consequently lost? The Plan of Salvation.
And second, once a knowledge of the Plan of Salvation was lost, then "many covenants of the Lord" were lost (1 Nephi 13:26). The covenantal nature of baptism, the covenantal renewal in the sacrament, the covenant of marriage, and all other temple covenants were lost. With the understanding of covenants lost, so too was the oath and covenant of the priesthood. It is not as if a few, small, unimportant truths were lost. The very foundation of the gospel was somehow taken or fell away.
Third, once the very foundation of the gospel and its covenantal nature had disappeared, what was lost next? Only then were things "taken away from the book" (1 Nephi 13:28). With the foundation missing, the text and doctrine that was left needed to be justified and explained. Because of this, certain writings (like some texts in the Dead Sea Scrolls) were no longer seen as necessary, or became an embarrassment. For example, the Gospel of Barnabas (a very early Christian text) talks a lot about the physical resurrection of the Lord. That was standard doctrine until about the 3rd century, and then it became a point of contention. Why? Well it became a snag in the gospel fabric already made of tatters. If you believe that God is without a body and is now dwelling in the heavens, then you cannot have scripture that conflicts with that. You must take it out and remove it from the Bible. Things were removed and then even more was taken out to account for the holes. But often, the words of the Bible remained, but their meanings were shifted. Things can be taking out of an ancient text simply because their meanings are not preserved and become lost due to lack of memory when no one remembers what they originally meant.
Book of Mormon Central, "Were Plain and Precious Doctrines Lost? (1 Nephi 13:26)," KnoWhy 15 (January 20, 2016).