The false apostles boasted about their visions. If Paul wanted to he could boast all the more about this, but he does not want to boast about visions he has seen, for he knows that it is of no value. Yet at the same time he needs to get the Corinthians to see sense and stop being taken in by the false apostles. So he talks about the vision in an oblique way. The "man in Christ" is most probably Paul himself. At some point God had lifted him up to paradise itself.
So Paul could outmatch the false apostles in terms of visions, but he has no wish to do so. The only aspects of himself that he will boast about are his weaknesses. Why? Because it is here that the power of God's grace and love are shown most clearly. Our role in life is to display the splendour of the Lord.
So Paul tells of a time when God failed to answer his prayers! Now there are many speculations on what the "thorn in the flesh" was, the simple answer is that no one knows, so I am not going to join the speculations.
The thorn came from Satan, but was allowed by God in order to keep Paul from becoming conceited. See the humility here. Paul was aware of his weaknesses, but also knew God's strength. That is the mark of a true believer. Paul pleaded with God to take the thorn away, but God refused to do so.
So we see something about prayer here, but it is not that we should be weak and vague in prayer. There are times to be specific and to pray with authority, but ultimately if something is not going to be good, then I do not want God to give it to me, and I recognise that only God has perfect wisdom. So when I pray I am happy for God to overrule me, and mighty glad that He will do so. But there are two key lessons to learn here.
One is that some people talk as if we pray for the wrong things it can hinder the work of God. If we are praying out of a hard heart, then God may give us the thing we are asking for in order to bring us to our senses. But if I am praying for the wrong thing, but from good motives, then I trust that my Father in heaven will gently say no.
The second is that this is not a reason for weakly adding "if it be your will" to all our prayers. Sometimes we can be almost certain what God's will is, or at least have a pretty good idea. So often we should pray specifically and pray with authority fully expecting God to answer. We can do this with the confidence of children of God.
In this situation God was teaching Paul about His grace. His grace is sufficient to overcome all our weaknesses, and His grace is the foundation of our confidence. So Paul, unlike the false apostles, was happy to boast about his weaknesses and difficulties, for in these Christ's power was most clearly demonstrated.
May the power of Christ be demonstrated in our lives!