The basic reason for the "removal" of the Apochrypha from the canon is that the Jews never recognized them as scripture, they are not part of the Hebrew canon of the Old Testament. There were a lot of books written over the years, some of them were even held in quite high esteem. Take the Book of Jasher which is mentioned in a couple of places, like Joshua 10:13, it is basically treated as if it's written in the book of Jasher, you can trust it happened, yet still it is not inspired scripture.
Joshua 10:13 KJV And the sun stood still, and the moon stayed, until the people had avenged themselves upon their enemies. Is not this written in the book of Jasher ? So the sun stood still in the midst of heaven, and hasted not to go down about a whole day.
So what happened at the time of the Reformation is that the Protestants recognized that the Apochryphal books were not God-breathed, they were not written by an oracle of God. So we ended up with the same New Testament as the Catholics but the Hebrew Old Testament as canonical.
Now the major authority used to argue for the Apochryphal books being included is the Septuagint and that Jesus and the Apostles quoted from it. Note that though they both point to the Septuagint, the Catholics and Orthodox still have slightly different canons. But anyway, most of the Apochryphal books did not even exist when the Septuagint was translated, some, authorities seem to pretty well agree weren't even written until after Jesus. Yet, there seems to be this belief that Jesus was quoting from a Septuagint exactly like the Septuagint of today. That seems not to be the case, based on our best knowledge. So the Septuagint arguement seems to have a lot of holes in it.
Then there is the internal evidence. Some of the Apochryphal books are clearly not written by whom they claim to be written by. Now we don't know with certainty who wrote all the books of the bible, but if a particular author is claimed, it had better be correct to be scripture. There are also other mistakes in the books. And if you read them many just do not read as scripture. So external evidence for inclusion is weak at best, internal evidence is weak at best and when you consider external and internal evidence it's pretty clear in most cases that they aren't scripture.
That's just a small summary, there are of course entire books written on the subject and the arguements can get very detailed.
Marv