Paul goes into great detail to describe how he became closely acquainted with the 12 apostles much later in his Christian life. He spent three years in Damascus before meeting Peter, with an undefined amount of time spent in Arabia before moving to Damascus. That time lapse seems to fit into Acts 9:25-26. Paul was introduced to the Jerusalem Christians by Barnabas, and then was a house-guest of Peter's for fifteen days. He met James (Jesus' brother), but not the other ten apostles. Then Paul left Jerusalem and went to live in Syria and Cilicia. The Christian churches in southern Israel had never met him, and would not have recognized him by face. They only knew about Paul by reputation, and gave thanks to God for his salvation.
Why go into so much detail?
It was because Paul's apostleship was under attack. The Jewish cultists who had infiltrated the Galatian congregation were claiming that Paul was just a junior apostle. A "small-a" apostle. They did this, so that the Galatians would stop listening to Paul, and listen to them instead. So Paul needed to defend his credibility. The Lord had given Paul his ministry from the very beginning of Paul's conversion (see Acts 9:20-22). Paul was also healing and casting out evil spirits from the very beginning. Paul was the 13th apostle.
What can we learn from this section?
It is right to defend your credibility when attacked. Punching-bag pacifism is not scriptural. Standing by silently while someone falsely tears your good name apart is a dereliction of your witness. If people lose faith in you, they won't listen to your witness. Satan attacks the gospel by ruthlessly smearing the spokesman. You must deserve a good name in reality, and defend your good name in the world.
When Paul wrote, Jesus Christ wrote. Religious liberals commonly smear Paul as a misguided systematizer who led us away from the "simplicity" of Jesus Himself. An example of this is so-called Christian feminism, which often portrays Paul as a well-intentioned man blinded by his culture at best, and a woman-hater at worst. This is rubbish. Paul was Christ's personal representative. Paul wrote these letters, and Jesus Christ was writing them through Paul.
Do not let anyone steal your good name. It is an essential part of having a ministry in the world.