There is a waiting period between the end of a battle and the moment when you can claim the reward. Once a reward is available, the Claim button will become clickable.
It is possible to claim a reward earlier before the end of the waiting period, but you will receive a lower reward as a percentage of the remaining waiting period. For example, after the end of a battle, your waiting period to claim your reward is 5 days. 1 day has passed and you want to claim the reward. At this point in time, you can only take 20% of your reward, and the rest of the reward will be burned away irretrievably.
The rewards will be credited to your in-game wallet. You can withdraw them to an external wallet that supports BEP-20 tokens any time.