"If I proclaim the gospel, this gives me no ground for boasting, for an obligation is laid on me"
For Paul, the gospel (or the good news) has to do with both the gospel of God (c.f., Rom 1:1) and the gospel of Christ. The gospel of God means the good news about God and the good news that God declared through Abraham and other prophets.
The good news of Christ is the gospel Jesus proclaimed. Namely, he proclaimed the gospel of God. The gospel of Christ also means the good news about Christ because he did good things for God and people. Christ's love and grace are also important. Paul believes that Christ demonstrated God's righteousness through faith. In other words, he risked his life and he was crucified because of his bold proclamation of God's good news, not Rome's good news or Rome's rule.
Therefore, Paul proclaims the good news of God through Christ. That is, he emphasizes Christ crucified (1 Cor 2:2) and his faithfulness and his grace. The gospel is the power of God for salvation to everyone who has faith. Paul is not ashamed of the gospel. Preaching is his duty to make this truth and power of God effective to all. It is not for boasting. He believes his God-given calling is to proclaim the good news of God through Jesus.
For this job, he does not want to burden anyone with financial things. He wants to live a simple life, so that the gospel may be proclaimed to the poor and weak who cannot pay him at all. An obligation is laid on him because he was called by God to proclaim the gospel to the Gentiles.