Here is a section of the statement addressing the relationship between the Book of Abraham and the text on the recovered fragments, including the non-Abraham-related text around Facsimile 1, which is often cited as proof that the Book of Abraham is a fraud (references omitted):
Of course, the fragments do not have to be as old as Abraham for the book of Abraham and its illustrations to be authentic. Ancient records are often transmitted as copies or as copies of copies. The record of Abraham could have been edited or redacted by later writers much as the Book of Mormon prophet-historians Mormon and Moroni revised the writings of earlier peoples. Moreover, documents initially composed for one context can be repackaged for another context or purpose. Illustrations once connected with Abraham could have either drifted or been dislodged from their original context and reinterpreted hundreds of years later in terms of burial practices in a later period of Egyptian history. The opposite could also be true: illustrations with no clear connection to Abraham anciently could, by revelation, shed light on the life and teachings of this prophetic figure.I personally don't like the catalysis theory and don't think it is necessary to deal with some of the issues in the Book of Abraham, but it is one of several commonly discussed possibilities.
Some have assumed that the hieroglyphs adjacent to and surrounding facsimile 1 must be a source for the text of the book of Abraham. But this claim rests on the assumption that a vignette and its adjacent text must be associated in meaning. In fact, it was not uncommon for ancient Egyptian vignettes to be placed some distance from their associated commentary.
Neither the Lord nor Joseph Smith explained the process of translation of the book of Abraham, but some insight can be gained from the Lord’s instructions to Joseph regarding translation. In April 1829, Joseph received a revelation for Oliver Cowdery that taught that both intellectual work and revelation were essential to translating sacred records. It was necessary to “study it out in your mind” and then seek spiritual confirmation. Records indicate that Joseph and others studied the papyri and that close observers also believed that the translation came by revelation. As John Whitmer observed, “Joseph the Seer saw these Record[s] and by the revelation of Jesus Christ could translate these records.”
It is likely futile to assess Joseph’s ability to translate papyri when we now have only a fraction of the papyri he had in his possession. Eyewitnesses spoke of “a long roll” or multiple “rolls” of papyrus. Since only fragments survive, it is likely that much of the papyri accessible to Joseph when he translated the book of Abraham is not among these fragments. The loss of a significant portion of the papyri means the relationship of the papyri to the published text cannot be settled conclusively by reference to the papyri.
Alternatively, Joseph’s study of the papyri may have led to a revelation about key events and teachings in the life of Abraham, much as he had earlier received a revelation about the life of Moses while studying the Bible. This view assumes a broader definition of the words translator and translation. According to this view, Joseph’s translation was not a literal rendering of the papyri as a conventional translation would be. Rather, the physical artifacts provided an occasion for meditation, reflection, and revelation. They catalyzed a process whereby God gave to Joseph Smith a revelation about the life of Abraham, even if that revelation did not directly correlate to the characters on the papyri.
The article also discusses evidence for the authenticity of the text, mentioning a few of the interesting finds where the text as well as Joseph's comments on the facsimiles have been shown to have surprising and interesting support. The "hits" mentioned are far from exhaustive but should be sufficient to give pause to those who have been told that the Book of Abraham is a complete fabrication without a shred of evidence to support it.
Related resources: LDSFAQ page on the Book of Abraham, Part 1, Part 2, Part 3: Evidences for the Book of Abraham.
Update, July 16, 2014: A related article is "New Gospel Topics Essay: 'Translation and Historicity of the Book of Abraham.'" In addition to some useful points about the evidence for the Book of Abraham and other aspects of the Gospel Topics article, there is an intriguing observation about some of the sources relied on in the Church's statement:
As an aside, I also find it significant that this essay cited material from both “classic FARMS” publications, such as Hugh Nibley’s The Message of the Joseph Smith Papyri: An Egyptian Endowment, as well as Interpreter: A Journal of Mormon Scripture. This would seem to indicate, I believe, that the claim, made by some, that the Church is trying to distance itself from these materials should be accepted with a bit of skepticism.If you're not a fan of The Mormon Interpreter, head over there and start digging in. Useful and intelligent material for LDS readers and investigators.