In this post I will try to explain how the taxonomy field in SharePoint 2013 auto-completes the terms as you start typing. In order to figure this out enable developer tool and view the asynchronous request triggered as you type in the taxonomy field

An asynchronous call to the Taxonomy Internal service is triggered and try to get the suggestions based on the user input.
by looking at the asynchronous calls the below is the request triggered

 also in the below snapshot is the parameters sent along with the request

To check complete reference of the taxonomy internal service documentation check the link below

It's 2010 version but it's still valid but I wonder why I can't find link to 2013 in  other versions drop down