Who was the author? The Holy Spirit inspired Moses (and perhaps a later editor or two) to record the inerrant words of the book of Numbers.
What is the book? Numbers continues the historical narrative of the people of Israel moving from Sinai to Canaan, although their unfaithfulness forced them to spend 40 years in the desert before returning to Canaan's borders. (There is surprisingly little about those 40 years in the book, however.) The book gets the name by which we know it from the censuses recorded in chapters 1 and 26, but the Hebrew name for the book, which is translated as "In the wilderness", in some ways is a better title.
Where was it written? Portions of the book may well have been written as the people wandered around the Sinai peninsula's desert, but certainly the concluding chapters of the books narrative at least were recorded on the east side of the Jordan River across from Canaan.
When was it written? A usual dating scheme has the people wandering in the desert from 1446-1406 B.C., and so we would put the writing of Numbers during that time, with at least its later portions coming at the end of the period.
Why? The events of Numbers are an important part of the salvation history of God's people. They sinned against Him and were forgiven, just as in our time we sin against Him and are forgiven by grace through faith in Jesus Christ.
How? Numbers shows us God's grace and redemption in a "number" of different ways, including the "numbers" that give the book the name by which we know it. From the 70 that went into Egypt at the end of Genesis to the 601,730 fighting men alone that stood poised to enter Canaan at Numbers end, we see such things as how God blessed His people despite their unfaithfulness, and we can think ahead to the perfect "number" of those saved in the Church and "number" ourselves among them.
For further reading on the book of Numbers:
- Keil, C.F. and F. Delitzsch. Biblical Commentary on the Old Testament, Volume III, The Pentateuch, translated by James Martin and published in one volume with the other two on the Pentateuch. Grand Rapids, Michigan: William B. Eerdmans Publishing Company, reprinted May 1986. (The section dealing with Numbers, "The Fourth Book of Moses", runs 268 pages. After my study Bible, this is what I turn to next, but it is a somewhat harder to use more-scholarly commentary.)
- Kretzmann, Paul E. Popular Commentary of the Bible: The Old Testament, Volume I, The Historical Books of the Old Testament: Genesis to Esther. St. Louis: Concordia Publishing House, 1923. (Like our Grace library, I have this volume on my shelf, and, although I didn't pull it down at all while blogging on Numbers this time through, I would think you would find its 66 pages on Numbers somewhat helpful and generally accessible. The format runs the text in bold with the comments immediately following the relevant text, so you can in effect read the whole text and his conveniently-placed comments, if you like that format.)