It’s for the anchor text of the link. For example, if you have a link “Click here” that points to Google, then “Click here” will be displayed as the link text.
In addition to links, the plugin can also check images (you can enable/disable this in Settings -> Link Checker). Since the concept of link text doesn’t really apply to an image, it displays the text “Image” instead.
As for “(none)”, that shows up for links that have no anchor text. For example, if you’ve got an image link – e.g. a thumbnail that links to the full-view version – the plugin will display “(none)” in the link text column.