How does the fingerprint work?
In general they use pattern matching algorithms and hashes of various patterns in your finger. I wont pretend to know the exact mechanism, but I am fairly confident that your fingerprint cannot be extracted from stored data.
Why does Microsoft claim fingerprints are more secure than passwords?
This likely comes down to psychology rather than the actual mechanism behind verifying fingerprints.
If either the fingerprint or the password verification would have any known major flaws, I would like to think that someone would've fixed them.
However there are multiple non-technical problems with passwords. The average person wants to log into their computer quickly, and thus chooses a short and memorable password (not secure). Passwords can be susceptible to shoulder surfing. And worst of all I personally know people who write their password on their computer using a sticky note. None of these can be done with a fingerprint.
That said, fingerprints are not without their flaws either. However most problems that arise with them wont affect the average person. For example with expensive equipment fingerprints can be extracted from objects you touch. Furthermore, (a fact that hopefully doesn't concern many people) fingerprints have different protections by law, in the United States law enforcement cannot force you to give up any passwords, which is not the same for fingerprints.
The gist of it is that (from my understanding) you are told that fingerprints are more secure, as they are for most users. As for the few for whom its not enough, they should probably use both.