There are two ways:

1. The customer sees their own count. Their "Invite your Friends" referral popup shows a live count ("N friends") and, further down, lists the names of everyone who registered through their link.
2. Admins have a dedicated report. Go to Reports > Referred Users (menu item "Referred Users"). It lists every customer who registered through someone else's referral link, with Registration Date, First Checkout Date/ID, and a Referred By Customer column linking back to the referrer. Filter or search the table by the referrer's name to count how many people they referred.

There isn't a one-click "referral count" on the customer's own profile page in the admin, but the Referred Users report gives you the same information for every customer at once.