The official explanation for Pnut badges can be found on https://pnut.io/docs/resources/badges.
Badges are a way to show your support for Pnut. You have to change your profile preference to show your badge, for it to show up.
You can buy a badge for Pnut which unlocks some special features like File Storage, repost and follow E-mail notifications, and RSS feeds of your personal/unified streams for Pnut.
The intention behind the badge is to support Pnut directly and sustainably. The extra features are a thank-you. That is why above the minimum, you can choose to give more, and you have the option of displaying your badge on your profile.
For more, see the update at https://pnut.io/updates#2017-07-25.a.