Apple explains the Secure Enclave like so:

Touch ID doesn’t store any images of your fingerprint. It stores only a mathematical representation of your fingerprint. It isn’t possible for someone to reverse engineer your actual fingerprint image from this mathematical representation. The chip in your device also includes an advanced security architecture called the Secure Enclave which was developed to protect passcode and fingerprint data. Fingerprint data is encrypted and protected with a key available only to the Secure Enclave. Fingerprint data is used only by the Secure Enclave to verify that your fingerprint matches the enrolled fingerprint data. The Secure Enclave is walled off from the rest of the chip and the rest of iOS. Therefore, iOS and other apps never access your fingerprint data, it’s never stored on Apple servers, and it’s never backed up to iCloud or anywhere else. Only Touch ID uses it, and it can’t be used to match against other fingerprint databases.

Want to know more? Mike Ash digs deeper.