It refers to the coming of the New Covenant versus the Old Covenant. However, there is some debate about the precise meaning. Jesus obviously had the first century scribes and Pharisees in mind, but the question wants to know exactly what it was about them that Jesus had in mind. The parable of the wineskins makes the same point as the parable of the new patch on the old garment. Paul in Galatians makes a good application of this parable, that Christianity doesn't make Gentiles to first become Jews by following the Mosaic Law.
yet we know that a person is not justified by works of the law but through faith in Jesus Christ, so we also have believed in Christ Jesus, in order to be justified by faith in Christ and not by works of the law, because by works of the law no one will be justified.
(Gal. 2:16, ESV)
McGee makes the point.
Our Lord is saying this, “I haven’t come to sew patches on an old garment. I have come to present a new garment, something which is altogether new.” This was very radical. John summed it up in his gospel when he said, “For the law was given by Moses, but grace and truth came by Jesus Christ” (John 1:17).
McGee, J. V. (1997). Thru the Bible commentary (electronic ed., Vol. 4, p. 54). Thomas Nelson.
This Messianic Jewish commentary is one of the few that addresses this question of the wineskins:
Whereas in v. 16 Messianic faith has to be adapted to Judaism, here it is Judaism which must be adjusted to Messianic faith. If one tries to put new wine, Messianic faith, into old wineskins, traditional Judaism, the faith is lost and Judaism ruined. But if Judaism is freshly prepared, reconditioned so that it can accommodate trust in Yeshua the Messiah, both the faith and the renewed Judaism, Messianic Judaism, are preserved.
The concluding paragraph is:
The meaning of the figure is that the new wine of Messianic living cannot be poured into old religious forms if they remain rigid. But if the old religious forms become “fresh,” they can accommodate Yeshua. When “kainos” is rendered “new,” as in many translations, the implication seems to be that Judaism cannot possibly be a suitable framework for honoring Yeshua the Jewish Messiah—only the “new wineskin” of Gentilized Christianity will work. This is a peculiar conclusion, especially if it is recalled that Yeshua was speaking with his fellow Jews. As rendered here the point is that the only vessel which can hold the new wine of Messianic life in a Jewish setting is a properly renewed, restored, reconditioned and refreshed Judaism, such as Messianic Judaism was in the first century and aims to be now.
Stern, D. H. (1996). Jewish New Testament Commentary : a companion volume to the Jewish New Testament (electronic ed., Mt 9:17). Jewish New Testament Publications.
While John Piper uses the term professionalism, I would reword what he writes as career (old wine skins) versus calling (new wine skins).
The world sets the agenda of the professional man; God sets the agenda of the spiritual man. The strong wine of Jesus Christ explodes the wineskins of professionalism. There is an infinite difference between the pastor whose heart is set on being a professional and the pastor whose heart is set on being the aroma of Christ, the fragrance of death to some and eternal life to others (2 Cor. 2:15–16).
Piper, J. (2002). Brothers, we are not professionals: a plea to pastors for radical ministry (p. 3). Broadman & Holman Publishers.
Jon Courson puts it as the new versus tradition:
You can’t put new wine in hardened old wineskins because when the new wine begins to ferment, the old hardened structure can’t flex with it, causing the wineskin to burst and the new wine to be lost. When people try to put something of a new moving of the Lord into an old structure, they end up not only quenching the wine of the Spirit, but blowing apart the structure in the process.
Courson, J. (2003). Jon Courson’s Application Commentary (p. 323). Thomas Nelson.
When Jesus brought the Good News of the gospel, he brought new wine into the marketplace of ideas. As a teacher, he did not fit the rigid concepts of the past. He taught revolutionary, explosive ideas that shattered hundreds of years of traditions. If his new wine were to be preserved, new receptacles were required. People had to change.
Reapsome, J. (2008). 10 minutes a day with jesus: growing in your love for the savior. Baker Books.
MacEvilly in Gill's commentary even applies the parable strictly to Jesus' ministry with limited application to today other than application to new converts. Matt. 9:15 gives this some precedence.
The application of these similitudes is quite easy, and is meant by our Redeemer to justify His mode of acting in not subjecting His disciples at first to the rigours of penance, for which, in their present imperfect state, they were unfit. His disciples He compares to old garments and old bottles; an austere system of life, to new cloth and new wine; and He argues, that if His disciples were all at once subjected to austerities quite new to them, they might fall into despondency, and desert His service altogether. Austerities are reserved for the time when, after being disciplined in the school of perfection, they shall become strong in the fulness of the grace of God’s Holy Spirit.... The application is the same as the foregoing. Those who are lately converted, are unable to bear the heavy burdens for which the fulness of the grace of the Spirit will fit and strengthen them.
MacEvilly, J. (1898). An Exposition of the Gospels of Matthew and Mark (pp. 170–171). M. H. Gill & Son; Benziger Brothers.
CHRYSOSTOM. Here again He confirms what He has said by examples of common things; No man putteth a patch of undressed cloth into an old garment; for it taketh away its wholeness from, the garment, and the rent is made worse; which is to say, My disciples are not yet become strong, but have need of much consideration; they are not yet renewed by the Spirit. On men in such a state it is not behoveful to lay a burden of precepts. Herein He establishes a rule for His disciples, that they should receive with leniency disciples from out of the whole world.
Thomas Aquinas. (1841). Catena Aurea: Commentary on the Four Gospels, Collected out of the Works of the Fathers: St. Matthew (J. H. Newman, Ed.; Vol. 1, p. 343). John Henry Parker.
JEROME. Otherwise; By the old garment, and old skins, we must understand the Scribes and Pharisees; and by the piece of new cloth, and new wine, the Gospel precepts, which the Jews were not able to bear; so the rent was made worse. Something such the Galatians sought to do, to mix the precepts of the Law with the Gospel, and to put new wine into old skins. The word of the Gospel is therefore to be poured into the Apostles, rather than into the Scribes and Pharisees, who, corrupted by the traditions of the elders, were unable to preserve the purity of Christ’s precepts.
Thomas Aquinas. (1841). Catena Aurea: Commentary on the Four Gospels, Collected out of the Works of the Fathers: St. Matthew (J. H. Newman, Ed.; Vol. 1, p. 344). John Henry Parker.
RABANUS. The different comparisons all refer to the same thing, and yet are they different; the garment by which we are covered abroad signifies our good works, which we perform when we are abroad; the wine with which we are refreshed within is the fervor of faith and charity, which creates us anew within.
Thomas Aquinas. (1841). Catena Aurea: Commentary on the Four Gospels, Collected out of the Works of the Fathers: St. Matthew (J. H. Newman, Ed.; Vol. 1, pp. 345–346). John Henry Parker.
Warren Wiersbe puts the application of this verse in a wider context.
Second “discipleship” interlude (vv. 9–17). We have covered the call of Matthew in the first chapter of this study. We need only to comment on the four pictures of His ministry that Jesus gave in this message. As the Physician, He came to bring spiritual health to sick sinners. As the Bridegroom, He came to give spiritual joy. The Christian life is a feast, not a funeral. The illustration of the cloth reminds us that He came to bring spiritual wholeness; He did not come to “patch us up” and then let us fall apart. The image of the wineskins teaches that He gives spiritual fullness. Jewish religion was a worn-out wineskin that would burst if filled with the new wine of the Gospel. Jesus did not come to renovate Moses or even mix Law and grace. He came with new life!
Wiersbe, W. W. (1996). The Bible exposition commentary (Vol. 1, p. 35). Victor Books.
In both cases, the result is the same. But, besides its special lessons, the second simile is also intended to show how entirely false the view alluded to in the first simile was, that Christianity was only a piece of new cloth to mend the torn garment of the old theocracy.
Lange, J. P., & Schaff, P. (2008). A commentary on the Holy Scriptures: Matthew (p. 171). Logos Bible Software.