It's assumed he spent the time in prayer and study preparing for his future ministry.
There's no evidence Paul spent the three years in quiet contemplation. We just assume he did.
1. Paul's missionary work began in Damascus immediately after his conversion (Acts 9:19-22).
He proclaimed Jesus in the synagogues as the Son of God and promised Messiah.
The opposition stirred up by Paul's missionary work is an indication that his preaching was successful and resulted in a good number of Jewish believers in Jesus.
2. Paul went to Arabia to continue his missionary work in obedience to God's call (Gal 1:15-17).
Arabia (in the region of modern Jordan) was not just desert, but also a flourishing civilization made up of cities, sea ports and cultivated land.
In the cities of Nabatean kingdom, south of Damascus, there were synagogues that Paul could have used as an entry point.
Paul's conversion was also his calling to mission. He didn't suddenly become a missionary years after his conversion. Paul met the risen Christ. He was commanded to preach the gospel and go to the Gentiles. Three years of solitude in the desert does not fit.
Luke never says Paul went "into the desert," as John the Baptist and Jesus had done. Luke says he went "into Arabia" where there was both desert and civilization.
Paul relates his Arabian visit closely with his call to preach Christ among the Gentiles (Gal 1:15-17). The point he is making to the Galatians is that he began to discharge this call before he went up to Jerusalem to see the apostles. Therefore none could say it was they or any human authority who commissioned him as an apostle to the Gentiles.
When his mission was complete in Arabia Paul returned to Damascus where the representative of King Aretas of the Nabateans sought to have Paul arrested (2 Cor 11:32-33). Why would Nabateans take action against Paul if all he had been doing in Arabia was prayerful contemplation?
Paul's mission in Arabia had stirred up trouble.
3. Therefore Paul's missionary career began immediately he was converted.
By the time Paul and Barnabas set off on what we call their "first missionary journey" (Acts 13) they were already seasoned missionaries who had seen both Jews and Gentiles come to faith, and churches established.
Why is this important? Missionary movements mobilize new disciples, like Paul, immediately for evangelism and church planting. They are action oriented. Their leaders learn their theology on their feet.