What is "Manuscript Evidence" and how is it useful? In several Apologetics articles (including one answer of mine on a different question), I see the statement that "there is overwhelming manuscript evidence to support the fact that the Bible has been transmitted accurately through time".  Often, these sites contain a chart like the following, with no explanation of why the number of copies, or the gap between the original written version and the earliest known manuscript matters.



So how does all of this prove anything?  Does it prove the Bible is the Word of God?  If not, what's the point?
"Manuscript Evidence", as the question implies, serves to provide evidence that the Bible has been transmitted accurately throughout time.  It does not prove that the Bible is God's word. 

The usefulness of knowing what manuscript evidence means, and what it tells us about the accurate transmission of Scripture through time is to address the statement that "The Bible has been changed many times over the years" - a common charge leveled by atheists and non-Christians.

The manuscript evidence argument stems from a branch of literary criticism, which seeks to identify and eliminate copying errors that occur over time.  Regardless of how careful people may be when copying texts, it is all but inevitable that errors will be made along the way.  The more often a piece of work is copied, the more likely an error will occur.

In layman's terms, the process of identifying and removing errors is done by comparing various copies of ancient works, and using a rigorous approach to determining what the original document actually contained.  All discrepancies are examined, and the original intent is deduced.

How well this works depends heavily on two factors, which are included in the chart in the question: 


The number of copies is useful because it gives a wider amount of copies to examine.  The more copies there are, the more likely there will be discrepancies.

This is good, because it helps to weed out the obvious errors - there will be fewer instances of errors because it's not as likely that various transcribers will make the same mistake.
Similarly, additions to the text, intentional alterations can be detected if they are present only in a limited number of texts.

The date between the original writing of the text and the earliest known manuscript is also critical.  Put simply, there is no way to compare copies of the text for manuscripts that do not exist, so if early mistakes occurred, resulting in discrepancies in later copies, it's impossible to ascertain which is more likely to be correct.

If the earliest known manuscript of a recorded event is within the lifetime of those who experienced the event, the argument is stronger, as there would be people alive at the time of the writing to correct or dispute inaccuracies.



A good example of how comparing discrepancies to ascertain original meaning can be found at Reasoning from the Scripture Ministries:


  Let us suppose we have five manuscript copies of an original document
  that no longer exists. Each of the manuscript copies are different.
  Our goal is to compare the manuscript copies and ascertain what the
  original must have said. Here are the five copies:
  
  
  Manuscript #1: Jesus Christ is the Savior of the whole worl.
  Manuscript #2: Christ Jesus is the Savior of the whole world.
  Manuscript #3: Jesus Christ s the Savior of the whole world.
  Manuscript #4: Jesus Christ is th Savior of the whle world.
  Manuscript #5: Jesus Christ is the Savor of the whole wrld.
  
  
  Could you, by comparing the manuscript copies, ascertain what the
  original document said with a high degree of certainty that you are
  correct? Of course you could.


When it comes to applying textual criticism to Scripture, the New Testament in particular has overwhelming manuscript evidence.  For the Old Testament, the evidence is not nearly as overwhelming, and the argument for trusting that they have been transmitted accurately through time rests more on the extraordinary care that was taken by the scribes.

Of course, the Bible (or more accurately, translations of the Bible) has been changed over the years, due to mistakes (such as the one that resulted in "The Wicked Bible").  Such mistakes resulted in revisions to Scripture, and it is possible that more revisions will come.  However, overwhelmingly, we can place a high level of confidence that the Bible that we have today is very close to the original manuscripts, and that it has been transferred accurately through time.  

So when it comes to answering the atheist's charge that the Bible has been changed over the years, the argument that we can't trust the Bible is based on a distorted definition of "has changed".  What they are trying to imply is that we can't trust the Bible because it's changed so much, but the fact is that we have such overwhelming manuscript evidence that we can be very confident in the fact that the meaning of the original manuscript wasn't lost. Spelling and copyist errors, there are truly very few verses that appear to have been "added" or that the common-sense reading is disputed.



The above is meant to be a Layman's answer. and is in no way a complete, or in-depth explanation of the concept, or all of the principles applied to textual criticism.  If you're interested, there are plenty of resources on the web discussing this topic.  For further reading, and a deeper understanding of Textual Criticism, the Wikipeda article is as good a place as any to start.