/> at the end of a tag indicates that it is not "opening" a tag. It's something that was breought in from XML to help standardize the HTML markup. Otherwise, anything interpreting the HTML needs to know which tags can have contents and which cannot.
There is nothing useful that can be put "inside" a <link> tag- you only want to set attributes on the tag itself, so the tag ends with /> as shorthand for <link rel="whatever"></link>. Previously, they just never had a closing tag at all- you would have <link rel="whatever"> and the browser was expected to know that <link> tags don't need to be closed.