Apple requires all apps that support in-app purchases to have a Restore Purchase button. It lets you transfer in-app purchases to other devices (such as a new iPhone). You can find it on the Aaptiv app by clicking the Profile tab, then the the settings icon (the gear symbol).
Clicking Restore Purchase will restore your account if you're on a device that shares the same Apple ID as your previous device. Clicking Restore Purchase will not reactivate your subscription if you've already canceled it, nor will it double charge you if you already have a subscription.