How can the translators of the New World Translation say that their version is near what was originally written? The New World Translation of the Bible says much concerning the original Bible writings. As none are known to exist, how can the translators of the New World Bible, JWs, say that anything is anywhere near what was originally  written?
Jehovah's Witnesses base their faith in the preservation of the Bible on the merits of scholarly studies of thousands of manuscripts, as well as on the reasoning that if the Bible is truly inspired of God, Jehovah will not allow its message to be lost (Isaiah 40:8). The New World Translation was translated using the same master texts that many other Bibles use.
Jehovah's Witnesses have published many articles about the subject. Some central resources on the subject are the Insight book and Appendix A of the NWT. The appendix states:

Not all copies of ancient Bible manuscripts contain identical wording. How, then, can we know what the original text contained?
The situation could be likened to that of a teacher who asks 100 students to copy a chapter of a book. Even if the original chapter was later lost, a comparison of the 100 copies would still reveal the original text. While each student might make some errors, it is highly unlikely that all the students would make exactly the same ones. Similarly, when scholars compare the thousands of fragments and copies of ancient Bible books available to them, they can detect copyist error and determine the original wording.
How confident can we be that the thoughts contained in the original Bible texts have been accurately transmitted to us? Commenting on the text of the Hebrew Scriptures, scholar William H. Green stated: “It may be safely said that no other work of antiquity has been so accurately transmitted.” Regarding the Christian Greek Scriptures, or so-called New Testament, Bible scholar F. F. Bruce wrote: “The evidence for our New Testament writings is ever so much greater than the evidence for many writings of classical authors, the authenticity of which no one dreams of  questioning.” He also said: “If the New Testament were a collection of secular writings, their authenticity would generally be regarded as beyond all doubt.”

Some sources from the Insight book can be found here and here.