Why did Paul go to Arabia?

But when it pleased God, who separated me from my mother's womb, and called me by his grace, To reveal his Son in me, that I might preach him among the heathen; immediately I conferred not with flesh and blood: Neither went I up to Jerusalem to them which were apostles before me; but I went into Arabia, and returned again unto Damascus. (Galatians 1:15, KJV)

Note that the KJV translation of Galatians 1:15 is correct in saying "in me," rather than "to me." Many Bibles give "to me" because of the "Road to Damascus" conversion found only in Acts of the Apostles, where Paul actually went first to Damascus and preached only to the Jews there (Acts 9:20: "And straightway he preached Christ in the synagogues, that he is the Son of God").

Arabia in the first century was a term for the land of the Nabateans, with its capital at Petrea, near the southern tip of the Dead Sea. Some theologians suppose Paul went there to preach. If so, he was unsuccessful as there is no record of any converts, and Paul never mentions the Nabateans again.

It is instructive that King Aretas of the Nabateans was at this time also king of Damascus, as a gift from Rome. A very likely scenario is that Paul went to Petrea to seek the king's approval for him to preach to Gentiles in Damascus. The king would have been unconcerned if Paul only preached to Jews, since Christianity was still thought of as a sect of Judaism, but could have ordered Paul's execution if he decided to disapprove of the conversion of pagans. This concern was real, as Paul's success was eventually his undoing, when the governor of Damascus attempted to arrest Paul:

In Damascus the governor under Aretas the king kept the city of the Damascenes with a garrison, desirous to apprehend me: And through a window in a basket was I let down by the wall, and escaped his hands. (2 Corinthians 11:33)

Galatians 1:15-18 (ESV)

But when he who had set me apart before I was born, and who called me by his grace, was pleased to reveal his Son to me, in order that I might preach him among the Gentiles, I did not immediately consult with anyone; nor did I go up to Jerusalem to those who were apostles before me, but I went away into Arabia, and returned again to Damascus. Then after three years I went up to Jerusalem to visit Cephas and remained with him fifteen days.

It is not clear why Paul went to Arabia. Even Luke did not mention this in Acts. We can find this account only in his letter to Galatians. We may conclude that Paul stayed sometime in Damascus after the conversion, preaching to the Jews and then went to Arabia, and after three years only Paul went to Jerusalem to speak with the Apostles face to face.

Some scholars believe that Paul went to Arabia to receive revelations from Jesus Christ.

From the reading of the text in Acts 9:26-28 one would get the impression that Saul went directly from his escape at Damascus to Jerusalem. By his own admission he clarified the fact in his letter to the Galatians (1:16-17) that he “conferred not with flesh and blood” nor consulted with the Apostles in Jerusalem; but the Lord’s choice for him was to go to Arabia to be trained in the school of the Spirit in order that he might receive greater revelations concerning the mysteries of the Gospel of the glorified Christ. (source)

Some scholars believe that Paul went to Arabia to preach the Gospel and this action proves that Paul received his revelations concerning the Gospel from Christ himself.

Paul thus wrote to tell them that after his conversion, he preached among the Gentiles for an extended amount of time before ever meeting with another apostle. Paul did not hurry off to Jerusalem to get instruction and approval from the Twelve. In defense of his apostolic credentials to the churches of Galatia, Paul mentioned his delayed journey to Jerusalem in order to emphasize (among other things) his genuine apostleship, whose message and authority came from Almighty God, and not from the twelve apostles, or any other person. (source)

Some again believe that Paul needed solitude because his life suddenly turned upside down.

Another point to consider is Paul’s world was turned upside-down in a moment’s time. Wasn’t he human, just like the rest of us? Wouldn’t he need to withdraw for awhile in solitude to more or less come to grips with the powerful upheaval that had disturbed the certainty of his righteous stand in the Law? Wouldn’t he have needed some inner clarity? Up to this point in his life, he had served the Law, and through the Law, he was convinced that the new Messianic Jews taught blasphemy when preaching that the Temple sacrifices had become obsolete in Christ. (source)