From your BudURL dashboard, check the box next to the BudURL link you want to delete, and navigate to the dropdown menu in the top left of the orange bar on your link manager to delete. You can also check the top-most box to select all links on one page and mass-delete. Please note: Take care when using this feature - bulk deletion is not reversible!
Please see the attached screencast for a demonstration.
Note: The bulk delete feature is not available in unpaid versions of BudURL.