When a butterfly reaches Level 5 and is fully levelled up, their special ability is unlocked. This ability will be displayed in the Flutterpedia page, and each time you tap on your butterfly it can be seen on the banner on the screen.
To use an ability, tap on the butterfly - then tap 'ready' on the ability icon. Once you have done so, your ability will be actioned and the cooldown period will begin before you can use that ability again. Some abilities also require an extra step (such as upgrade an attract flower etc.) where you may have to select a butterfly/flower etc. to apply your ability to.
For early butterflies in the game there is a passive ability that increases the amount of honeydew earned by all other butterflies in that set. For butterflies later in the game, their special ability is an action that can be used, and recharges over time!