What does it mean to say an original Greek manuscript did not survive? What does it mean when they say Greek manuscripts, written by the original authors, did not survive? I don't Understand this statement.
Without context more specific than "they say", it is difficult to know what was meant. However, such a statement is likely to be referring to no longer having the original copies of New Testament writings. Only copies of copies are now available.

This is to be expected as wear by use, environment conditions, and the ordinary misplacement and destruction of items. The same statement can be made for other documents of similar age.

Consider this image of one of the Dead Sea Scrolls (as a convenient example):


This is a document that was intentionally preserved without use in a reasonably friendly environment and isolated and hidden enough to avoid destruction by war, accidental fire, etc. Although centuries older than any New Testament documents would be, this still hints at why copying is the most reliable means of preserving information.