How is this significant for understanding the world of Paul?

Paul picks up on athletic imagery to address the Corinthians thought they had already "made it" spirituality. Now that they were part of the church, they kind of a free-ticket to do whatever they wanted; nothing would harm them. Instead of using this freedom to serve others they used it justify their rather selfish lifestyle. Paul warns them that the "game" is not up.

Do you not know that in a race all the runners run, but only one gets the prize? Run in such a way as to get the prize. Everyone who competes in the games goes into strict training. They do it to get a crown that will not last; but we do it to get a crown that will last forever. (1 Corinthians, chapter 9:24-25)

How is Paul remixing his culture

At some level, Paul is simply employing a common image from his day to remind Christians to take their faith seriously. The race is not over and they cannot act as if their actions do no matter. He also reminds Christians that the prize is not simply an earthly one, but an eternal one.

Paul does tweak the metaphor a bit in that the way to run the Christian race is not simply to compete individually but to serve others. In the broader section (chapters 9 and 10), Paul makes two points.

• Paul describes how he uses his freedom to work as a laborer for the Gospel to save as many as possible. Paul does not employ the race imagery to imply that one only person or a very limited number will be saved.
• Paul describes how our actions are not limited by our freedom in Christ, but by the consciences of others. As Paul concludes:

"Everything is permissible"—but not everything is beneficial. "Everything is permissible"—but not everything is constructive. Nobody should seek his own good, but the good of others. (Chapter 10:23- 24)

Even if we are runners, our running cannot mean trampling over others!  As runners we know that we have a prize in store, but that the race is not over.