Why did the athletes in Paul's analogy strive to win?
The athletes in Paul's analogy strove to win for a perishable wreath—a temporary prize that would dry up and be fit for fire before they reached home. Paul contrasts this with believers, who should strive far more for an eternal reward in the heavens that will never shrivel or die.
Les explains that Paul often used the analogy of Olympic athletes. Those athletes trained and strove to win only to receive a wreath that would be all dried up and fit for the fire before they even got home; they did not get gold. Paul's point is that if they did all of that for such a temporary reward, believers should strive much more for reward that is eternal in the heavens, not something that will shrivel up and die. The context also ties reward to the believer's fruitful life: salvation is based on faith, but works result in reward laid up and waiting, to be enjoyed for all eternity.