If you need to delete a brand from My Brands, click the checkbox next to the brand name, then select "more actions". You'll get the options to enable, disable, reload, or remove the brand from your list.

Note that deleting the brand from My Brands will not delete the brand from your store.  This is intentional in case you want to update the brand manually or need to keep the products for other reasons.  To permanently delete the brand please go to the Shopify admin and delete the products using the Products tab.