Does anyone actually know how does touch id actually authenticates with apple ids as well? As in how does the login process for the App Store works?

As far as I'm aware, your iPhone stores your Apple ID credentials in a secure place in system memory, with a good level of encryption. When a user is asked to authorize a purchase, it will ask for your fingerprint. When this is read and is authorized with the information already kept about your print in the sealed off section in the A7 processor, the phone is given authorization to use your credentials to authorize a purchase on the store. Nothing at all to do with your print is ever sent over for use in the OS itself, only a simple 'Yes' or 'No' as to weather the given information matches up.