Using a OAuth token with the right scope, one of the below:
channel_subscriptions channel_check_subscription user_subscriptions
and polling the correct endpoint, for example /channels//subscriptions you can lookup the name/id of the most recent subscriber then you can hit the /users/ endpoint for more information
or gather the sub information via webhooks.
For more information: Twitch API | Twitch Developers and Twitch API | Twitch Developers