Experience won through battles is stored in the Claim EXP button, in the top right corner of the page
As claiming experience costs a gas fee for the transaction, it is recommended to only claim your experience for a character if it will push them to the next milestone if the character in question is under level 41.
Past level 41, it becomes beneficial to claim their experience every level thereafter as the boost in power will result in more SKILL gained through fight payouts.