It is normal for Paul to follow his initial greeting with a prayer of thanksgiving for the recipient church. Here he gives thanks, not just for the Romans’ faith but because it is widely known. When Paul talks about the world, he means the known world of the day or Roman Empire (v8). Paul himself is constantly talking about them and their faith, giving thanks for them to God (v9). And specifically, he expresses a desire to come and visit them (v10).
The reason Paul wishes to see them is that he desires to pass on to them a spiritual gift. Given that he has already described his own apostleship as a gift, I would presume there is a link here. He wants them to benefit from his apostleship. This is not about him passing on a gift through physical contact, where they catch the fire in someway as he lays hands on them. It’s rather that as a Jew with a calling to Gentiles, he has something to offer a church that includes both Jews and Gentiles together. The gift is his ability to teach and to speak into their situation (v11). He expects mutual benefit, they will encourage one another (v12).
Verses 13-15 reinforce the reason for his desire to come to them. This is something he has sought to do for a long time but has consistently been met with obstacles. Indeed, it will only be as a prisoner that he is eventually able to visit Rome. His aim is to “reap a harvest”, to see people brought into a living relationship with Christ in Rome reflecting his specific calling to Gentiles. Perhaps, and this would be reflected in the letter, his concern is to see the cross cultural ministry to Gentiles and the inclusion not just of non Jews but those regarded as weak and uncultured by Greek standards into the church and their discipleship (v13). As their apostle he has a sense of obligation. This is to all Gentiles regardless of class, status, intellectual capacity both those who consider themselves among the Greeks, the wise and taught and those seen as “barbarians”, uncivilised. Note the reference here to both “wise” and “foolish” echoing 1 Corinthians (v15).