If you've added your website to Ahrefs' Dashboard, you'll be able to see the number of referring domains you have:
However, when you click through to Site Explorer, you'll see that the number of referring domains is different:
The numbers on your Dashboard is taken from our Live Index. However, note that these numbers are not real-time, and they are only updated once a day.
Site Explorer Overview
The numbers on Site Explorer > Overview is also taken from the Live Index. These numbers are the latest value, which you can also see if you visit the Referring Domains report: