The numbers in parentheses, following the ship names, indicate the number of times the vessel is mentioned in the ESCN Database.
Early South Carolina Newspapers Database is/was strange beast. It was a database of entries from eight 18th century Charleston newspapers, and was based on a print version of the same data. The database could direct you to mentions of individuals, ships, slave references, advertisements, and organizations in specific issues of these eight different Charleston newspapers. Unfortunately, however, when its creator died in November 2010, the electronic database died with him.
You can still search the print volumes, one at a time, through the ship names section, in a library near you. You'll need to search every volume, because there's no way to know where to find the ship name you're seeking. Only then will you find the newspaper title and publication date. Of course, you need to be near a library that holds all of the available volumes. Even then, you'll only get references to South Carolina newspapers. You won't get the content from the newspapers themselves, so you'll still need to track down the specific SC newspapers to get the actual mentions of the vessels you seek.
A helpful ShipIndex.org user has told me that Accessible Archives' South Carolina Newspapers Collection is a much better alternative for finding this information. For a fee, Accessible Archives offers access to the original early SC newspapers online. If you've found the citation from the print version of ESCN, you can then locate it through Accessible Archives' database. Or, you can search for the ship name in Accessible Archives, and most likely find the citations that were also found during the creation of ESCN. (Of course, if the ship is named "Mary", you'll have a challenge on your hands looking through all the times the word "Mary" appears in the newspapers...)