As the LDS curriculum shifts from the Old Testament to the New Testament, there's a gap of five centuries between the post-Exile situation depicted in Ezra and Nehemiah and the New Testament writings of the first century. The Bible doesn't do much to fill that gap — at least the LDS Bible and others that omit the Apocrypha. I'm going to summarize the entry "Apocrypha" in the Oxford Dictionary of the Bible, then add a few comments about the LDS view of the Apocrypha and other non-canonical writings, including D&C 91.
So here's what the "Apocrypha" entry says:
After the Fall of Jerusalem (70 BCE) the future of Judaism was maintained by rabbis of the Pharasaic tradition. They accepted as authoritative the twenty-four books of the Hebrew Scriptures but rejected a number of Jewish works which were used in Alexandria and which are known to us in [manuscripts] of the [Septuagint] and called the Apocrypha (Greek for "things hidden away"). Being composed after the time of Ezra, when prophecy was held to have ceased, these Greek works, even if originally composed in Hebrew (e.g., 1 Maccabees), were unacceptable. By and large, Christians accepted the longer list ....
So the Christian tradition has been, on the whole, friendly toward the Apocrypha. A couple of other notes from the Oxford Dictionary entry:
The books of the Apocrypha are called deuterocanonical (= at second-level) by Roman Catholics, to distinguish them from protocanonical (= first-level) books, but they are regarded at authoritative and included at appropriate places within the body of the [Old Testament]. ... The exact extent of the Apocrypha is not universally agreed, and some of the books are known by different titles.
The LDS Bible Dictionary has a nice entry on the Apocrypha, calling them "valuable as forming a link connecting the Old and New Testaments" and terming them "useful reading." It also paraphrases D&C 91 (which directed Joseph not to "translate" the Apocrypha as part of his project rewriting selected texts from the Bible) in saying that "the contents [of the Apocrypha] are mostly correct, but with many interpolations by man."
A final source is James C. VanderKam's An Introduction to Early Judaism. VanderKam finds the usual terms Apocrypha and Pseudepigrapha too unclear (there is scholarly disagreement over which texts fall in which categories). Instead, he refers to the whole corpus as Second Temple literature, which he extends to include Old Testament writings from that period (such as Chronicles, Ezra, Nehemiah, Daniel, Haggai, Zechariah, and Malachi), as well as the writings of Philo and Josephus, and the Dead Sea Scrolls. I don't think you can get a better short introduction to the whole topic than VanderKam's book.
What's in it for an LDS reader?
There are a couple of reasons an LDS reader might find the Apocrypha interesting in light of the encouragement given by the LDS Bible Dictionary. First, it helps situate the New Testament. New Testament writers were familiar with the Apocrypha (which were included in the Greek Septuagint, the Bible of the early Christians). Themes that appear in the Apocrypha — such as angels, martyrs, and apocalypse as a genre — also appear in the New Testament. The Old Testament closes with Babylonians and Persians controlling the destiny of the Jews; the New Testment opens with Greek culture and Roman power on full display. The Apocrypha, with appropriate commentary, bridges that gap.
A second reason an LDS reader might find the Apocrypha interesting is that there are echoes of some texts in the Book of Mormon. It is hard to read the war narratives of 1 Maccabees without thinking about the war narratives in the last third of Alma. There is also the second chapter of 2 Maccabees, which recounts how the author abridged a five-volume work of Jason of Cyrene into a single volume and why he did so:
[W]e have aimed to please those who wish to read, to make it easy for those who are inclined to memorize, and to profit all readers. For us who have undertaken the toil of abbreviating, it is no light matter but calls for sweat and loss of sleep .... Nevertheless, to secure the gratitude of many we will gladly endure the uncomfortable toil, leaving the responsibility for exact details to the compiler, while devoting our effort to arriving at the outlines of the condensation.
Other posts and articles
For more discussion of the relation between the Book of Mormon and the Apocrypha, read Hugh Nibley's article "The Apocrypha and the Book of Mormon," originally published as Chapter 16 of his book An Approach to the Book of Mormon. A quick way to fill the gap is Kevin Barney's recent post at BCC, "Intertestamental Period.
I should mention the helpful LDS book that came out in 2006, Between the Testaments: From Malachi to Matthew, by S. Kent Brown and Richard Holzapfel. I wrote a couple of posts on the book: "Mind the Gap" and "Between the Testaments."