Paul is burdened because the true and living God is not being worshipped. This Scripture says that "Paul's' spirit was being provoked within him." The One worthy of ALL glory and praise is not receiving it. The city was full of all sorts of idols, perhaps as many as 20,000.
Paul is also burdened for the lost people around Him. That is why he was reasoning with the Jews, the Gentiles, those in the market place and anyone who happened to be present. Sharing Jesus Christ with them was of supreme importance to Paul. He shares the depth of his burdened heart with us as well in Romans 9:2-3: "... that I have great sorrow and unceasing grief in my heart. For I could wish that I myself were accursed, separated from Christ for the sake of my brethren, my kinsmen according to the flesh..."
This really challenges me! How burdened am I truly about the lost people around me? Does my compassion for them take priority over everything else? Do I weep for them? Do I pray constantly for them? Do I beg God for opportunities to open the door for me to share the sweet message of the gospel?
Lord, let me be burdened because You are the true and living God and You are not being worshipped by the whole earth as You deserve. Also, burden me with the condition of the lost that are around me everyday and find me faithful to pray for opportunities and boldness to take those opportunities.