Why were books left out of the bible and what do those books say that made them get left out?

THE SHORT ANSWER: The books that are in the bible underwent rigorous scrutiny for about 300 years. The ones that were not universally accepted as authentic by the churches of that era.

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It'd help, maybe to understand how the books were chosen. 

The few hundred years that followed Jesus' life there came to be quite a few christian churches that existed. Each of these churches accepted certain letters and books as "inspired by God." However, some churches accepted a hand full of books or letters that another church didn't. Likewise other churches accepted some that other churches didn't. It's important to know that most agreed, especially on the Jewish scriptures. What were in question were what they called, "the memoirs of the apostles." basically it was the writings of the apostles that were in question. 

There was no set list of accepted God-inspired books early on. Lots of letters and books were being circulated throughout the churches that were bogus, not written by apostles, or even made up. It became clear that something needed to be done. You can imagine the kind of confusion that could arise when each church would choose it's own scripture.

It became clear within a hundred years of Jesus that there needed to be a list of the books and letters that were UNIVERSALLY accepted by all churches. This was a brilliant way to do it because, it didn't force any new letters or books into the accepted collection. Instead it eliminated books that churches couldn't agree on. 

Here's how I would illustrate it. Lets say we have six Books. We'll name them A, B, C, D, E, and F
Now lets say we have three churches that use different combinations of books each. 

Church 1 uses Books A, D, E, and F
Church 2 Uses Books A, D, E, and F
Church 3 Uses Books A, B, C, D, E, and F

So if we made a list of the books that are universally accepted by all churches it would include only  A, D, E and F. It would exclude B, and C. 

Ok, so here's where it gets a little confusing. Some people will tell you that there are criteria that were used to determine which books were accepted... Some call these criteria the five laws of Canonicity. However, it's not quite true to say that the books were chosen based on the these criteria. 

It was more like these criteria were noticed along the way. As the church continued the discussion about which books were inspired, they developed these criteria to help guide the way. 

So the criteria are:

Was the book written by a prophet of God?
Was the writer confirmed by acts of God?
Does the message tell the truth about God?
Did it come with the power of God?
Was it accepted by God's people?

Over a period of a few hundred years the books slowly got sorted out, 

Archeology has reveled through the discovery of a fragment, that there was already a similar list to the modern one, in place by 200 AD. So we know that even though the discussion went on, the major books of the New Testament were already universally accepted between 100-200.

by the third century, there were very few differences between what they accepted and what is in the modern table of contents of your bible. 

It was kind of a known mission, to unify the list of Universally accepted "inspired books" for a few hundred years. 

Although it was eventually officially ratified by the church leadership, it was actually the church as a whole that gradually accepted these books as inspired. 

So in the 390s the list of biblical books was finally published and has stood since. 

It's not hard to find the other books that were left out of the canon of scripture. Some of them are quite interesting. Some of the ones that were used in the early church were The Gospel of Thomas, the Shepherd of Hermas, the Apocalypse of Peter, and the Epistle of Barnabas. 

The church ultimately determined that these extra books were either not inspired by God, and therefore did not belong in the bible.