The Catholic church created the concept based on misunderstood Bible verses and other non-Bible books, and declared it an official doctrine.
Protestants are opposed to the whole idea, as unbiblical. And I agree with the Protestants.
Here are some of the references that were the source of confusion for the Catholic church (the rest of the list is in the link below):
Jesus said: 28 Truly I tell you, people can be forgiven all their sins and every slander they utter, 29 but whoever blasphemes against the Holy Spirit will never be forgiven; they are guilty of an eternal sin.”
30 He said this because they were saying, “He has an impure spirit.”
Therefore, I tell you, people will be forgiven every sin and blasphemy, but the blasphemy against the Spirit will not be forgiven. Whoever speaks a word against the Son of Man, it will be forgiven him; but whoever speaks against the Holy Spirit, it will not be forgiven him, either in this age or in the one to come.
1 John 5:16-17
“If anyone sees a fellow believer committing a sin that doesn’t lead to death, he should ask, and God will give life to him — to those who commit sin that doesn’t lead to death. There is sin that leads to death. I am not saying he should pray about that. All unrighteousness is sin, and there is sin that doesn’t lead to death.”
The Catholic Church Point of View
The above references completely confused the Catholic church, and drove it to look for more references supporting their purgatory idea, both in the Bible and in other books! The one verse that drove the pope to make it official was, Matthew 12:32: "but whoever speaks against the Holy Spirit, it will not be forgiven him, either in this age or in the one to come." Their reasoning: "If this sin cannot be forgiven after death, it follows that there are other sins which can be, and this must be in purgatory". But this is so misunderstood, again by a bunch of people not filled with the Holy Spirit, and yet making big decisions.
The confusion resulted in the church making up their own list of "death sins" and "not death sins". Example, "lying" is not a death sin, "killing" is a death sin, "adultery" is a death sin, and this is why Mary is their perpetual virgin role-model. To resolve this salvation dilemma, they introduced "Purgatory". According to Catholic teaching, because the sin that leads to death is not one we can pray about and get forgiven, the Catholic church advised a place of suffering that people go to for a period of time for redemption, then on to heaven from there. A place similar to hell, but not as painful - purgatory. The Catholic church declared the purgatory doctrine around the 12th century based on some misunderstood verses and also referencing a non-biblical book called "Maccabees", an old Jewish book. This is completely unbiblical. It's another human doctrine to please people; as in: "Don't worry, the church got your eternity covered, if you commit death sins, you just swing by Purgatory and get slapped around some, then on to heaven. Not so bad, keep donations coming." Again here the Catholic church over-extended their power of the "bounding and loosening" in Matthew 18:15-19.
Here's an internet link listing the references that support Purgatory. I don't see anything here that supports Purgatory. Only misunderstood verses that they twisted in their direction. The link also references the unbiblical books of Maccabees twice, which that alone discredits the whole doctrine:
The Explanation Why It's Not Biblical
The first observation is that Jesus clearly said "only one sin is unforgivable", which is the same as the "death sin" in 1 John 5:16-17. There is no list of unforgivable / death sins; only one.
There was nothing in the above verses that alluded to some sins to be forgiven in this age and some in the next. It would be a stretch to try to read it that way. The reading more inline with the whole scripture would be that all other sins are forgivable in this age, by the blood of Jesus. To take a stretch of an explanation and turn it into a doctrine would be obnoxious and a clear indication of lack of the Holy Spirit within them. Why would the pope and his bunch want to go to such extremes to say that the rest of the forgivable sins are not all forgivable in this age by the blood of Jesus? That you still need to suffer some to get the full pardon. As in, "you can't just get off so easily after sinning all these horrible sins." A clear lack of knowledge of God! The pope and his assistants did not seek God. They were more filled with their own arrogance, public stature, and earthly politics than seeking God.
The fact that this verse from Jesus, "will not be forgiven either in this age or in the one to come." is not fully understood, does not mean we should build a new doctrine based on it. There are many parts in the Bible still not fully understood, and God will reveal them to us in His time. Some people spend a lifetime trying to figure out one misunderstood verse, with little to no knowledge of the rest of the Bible. The internet is full of articles with big words trying hard to make Purgatory fit. But the simple plain words answer is, it does not fit.
With wisdom from God, if I was to take a shot at the meaning of that verse from Jesus, "will not be forgiven either in this age or in the one to come." I would have a couple of possible answers:
Lucifer was an angel in heaven, and he committed the unforgivable sin of the spirit when he declared himself god. Then God cast him to "eternal" hell with no chance of repentance. People will have free will in heaven. Although we'd become incapable of sinning earthly sins, but the freewill keeps a potential to commit the unforgivable sin of the spirit just as Lucifer did. Although not likely, such a possibility would drive Jesus to include "the next age" in His statement.
The "next age" could also mean after the 1st death, but before the 2nd death. Which is somewhere in the millennial before, during, or after the Jesus thousand-year rule on earth in the end times. The Bible gave some information about this millennial period, but not the whole clear picture. As described in the book of Revelation, during that period some dead people would be brought back to life to be given their first chance at believing in Jesus as the Son of God. Notice I said "first chance". Correct, these are all the people who died before Christ (BC). As explained in some Revelation verses, God being gracious and fair, gives everybody a one chance to believe in Jesus. So if there are some among them that had committed the unforgivable sin, they would not be given that chance. But others with forgivable sins would be given a chance to repent.
Another interpretation could be, it was meant as a figure of speech - something like "not forgivable even when hell freezes-over". Because in the same verse in Mark 3:29, he uses the term "eternal" instead of "age to come". So the focus was on the severity of this sin, not the introduction of a new doctrine in 2 words.
We keep praying and seeking God. And He will reveal the answers to us at the right time.