urlBurp counts how many times the preview page for a short link was opened.
With an account
Create an account, then shorten links while you are signed in. The dashboard lists those links and the count for each one. A chart shows the last 30 days for your links together. Links you made before you had an account stay separate. They are not added to the dashboard later.
Without an account
After you shorten a link, this browser stores a private note of it. The home page can show those recent links and their counts. That note stays in this browser only. Clearing the browser's saved data removes it. The count on the server remains.
What the number means
The number goes up when someone is shown the preview. It does not tell you who they were. urlBurp stores a salted hash of the network address, not the address itself, along with the time, the referring page, and the browser type.