The short version is that a picture is taken of the fingerprint, in a given number of pixels. The sensor determines whether there's a ridge or a hollow at each position, or shades of gray between, and this varies from individual to individual. Just like any digitized image, the information is arranged in bytes, which represent each pixel. Now the information is stored in numeric format.
Then that information is analyzed, to identify a combination of features in the image that make it unique. It's essentially to identify patterns in your given n X n pixel scan. This algorithm is used to determine whether two fingerprints are from the same person. (Note, for example, that you cannot expect the finger to be placed in the scanner at precisely the same location every time, but it still has to be recognizable. Just using the pixel pattern from that initial scan is not enough for obtaining a unique and repeatable ID.)
Individual algorithms are most likely proprietary and not openly available, but this paper explains the basics.
Anyway, once you have that numeric sequence, it's not hard to imagine how it could replace a username and/or a password, in any normal crypto protocol.
Here's a very good survey on biometrics-based systems in general: http://www.cse.msu.edu/rgroups/biometrics/Publications/GeneralBiometrics/JainNandakumarRoss_50Years_PRL2016.pdf
This is another paper with concrete algorithms for alignment and bit extraction for fingerprints: http://www.merl.com/publications/docs/TR2010-003.pdf
Once you extract some bits from a fingerprints you should use a Fuzzy Extractor to generate a (reproducible) cryptographic key: https://en.wikipedia.org/wiki/Fuzzy_extractor