How does the information about recurring payment failure/success reach me?
Do I have to ping it myself or does BT send the response?
If it is the first case then when exactly should I check that? E.g when billing date is 1st August, then at what time it is done?
Full disclosure: I work at Braintree. If you have any further questions, feel free to contact support.
You can set up webhooks that notify you when the status of your subscriptions change. These webhooks are set up in the Braintree Control Panel and send POST requests to your server endpoints when needed.