What do catholics believe about purgatory?

That souls of those who weren't Internet trolls and other human garbage, end up there. So basically everyone who was not bad enough to end up in Hell. Then, after some time, a soul which resided in the Purgatory may go to Heaven and sing songs.

Possibly Paul didn't have enough of ink in his inkwell so he had to keep the story short.

You asked what Catholics believe about the Purgatory, so I answered.

Purgatory is a spiritual state where God shows us the merits of our sin by depriving us of His presence. As 1 Corinthians 3:15 states, "If the work is burned up, the builder will suffer loss; the builder will be saved, but only as through fire". This fire alludes to the agony we will face being without the mere presence of Our Lord. When we sin we hurt our relationship with God not by hurting Him in some way, but by hurting ourselves by turning away from Him and to ourselves to gain fulfillment. God wants nothing but for us to be fulfilled by coming into his presence and knowing the glory of who He is, since that is the grand mystery of our universe, but turning to ourselves in sin makes us say no to that salvation since it is a denial of Himself. In the world we live in we are so surrounded by God and His creation that we can not conceptualize what a world without Him is like, and the deprivation of even the smallest glimmer of that is the deepest kind of pain that we can bear.

Hell and Purgatory have the same purpose but different ends. Both Hell and Purgatory are separations from the presence of God, Hell being eternal and Purgatory being temporary.

When we die, our soul is separated from our bodies and we are judged in what is called the Particular Judgement, which discerns if we are worthy of Heaven or Hell. If we in our lives deny God, we go to Hell. If we are worthy of Heaven in some capacity, we are first cleansed by Purgatory in order to remove any shred of sinfulness that would destroy us when we would see the face of God in what is called the Beatific Vision.

Where is the scriptural support for purgatory? You said 1 Corinthians 3:15 “alludes to the agony we will face being without the mere presence of Our Lord.”
but it doesn’t say that.
1 Corinthians 3:15 KJV
[15] If any man's work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire.

Where does it say anything about the presence of God in this verse?

The wages of sin is death, not a deprivation of His presence,
Romans 6:23 KJV
[23] For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord.
James 1:15 KJV
[15] Then when lust hath conceived, it bringeth forth sin: and sin, when it is finished, bringeth forth death.
1 John 5:17 KJV
[17] All unrighteousness is sin: and there is a sin not unto death.

Sinners will not inherit the kingdom of Christ and God, but hell
Ephesians 5:5 KJV
[5] For this ye know, that no whoremonger, nor unclean person, nor covetous man, who is an idolater, hath any inheritance in the kingdom of Christ and of God.
Psalm 55:15 KJV
[15] Let death seize upon them, and let them go down quick into hell: for wickedness is in their dwellings, and among them.

God reigns in hell,
Psalm 50:3 KJV
[3] Our God shall come, and shall not keep silence: a fire shall devour before him, and it shall be very tempestuous round about him.
Psalm 68:2 KJV
[2] As smoke is driven away, so drive them away: as wax melteth before the fire, so let the wicked perish at the presence of God.
Proverbs 15:11 KJV
[11] Hell and destruction are before the Lord : how much more then the hearts of the children of men?
Revelation 14:10 KJV
[10] The same shall drink of the wine of the wrath of God, which is poured out without mixture into the cup of his indignation; and he shall be tormented with fire and brimstone in the presence of the holy angels, and in the presence of the Lamb:

Luke 16:19-31 KJV
[19] There was a certain rich man, which was clothed in purple and fine linen, and fared sumptuously every day:
[20] And there was a certain beggar named Lazarus, which was laid at his gate, full of sores,
[21] And desiring to be fed with the crumbs which fell from the rich man's table: moreover the dogs came and licked his sores.
[22] And it came to pass, that the beggar died, and was carried by the angels into Abraham's bosom: the rich man also died, and was buried;
[23] And in hell he lift up his eyes, being in torments, and seeth Abraham afar off, and Lazarus in his bosom.
[24] And he cried and said, Father Abraham, have mercy on me, and send Lazarus, that he may dip the tip of his finger in water, and cool my tongue; for I am tormented in this flame.
[25] But Abraham said, Son, remember that thou in thy lifetime receivedst thy good things, and likewise Lazarus evil things: but now he is comforted, and thou art tormented.
[26] And beside all this, between us and you there is a great gulf fixed: so that they which would pass from hence to you cannot; neither can they pass to us, that would come from thence.
[27] Then he said, I pray thee therefore, father, that thou wouldest send him to my father's house:
[28] For I have five brethren; that he may testify unto them, lest they also come into this place of torment.
[29] Abraham saith unto him, They have Moses and the prophets; let them hear them.
[30] And he said, Nay, father Abraham: but if one went unto them from the dead, they will repent.
[31] And he said unto him, If they hear not Moses and the prophets, neither will they be persuaded, though one rose from the dead.