It indicates that the content that follows is not inside of the tag without the reader needing to remember how HTML works. Tags should have either a self-closing slash, or a closing tag.
The third way of a bare tag is where the confusion comes from.
Contrary to <img /> or <br />, <div /> is necessarily a mistake or intentionally misleading. The unfamiliar reader should not stumble upon <div /> too often. <div /> is a bug. It's a bit like using misleading indentation in C-like programming languages. Yeah, it can happen, and is a source of bugs, but if the page is well written, the regularity of having everything closed, even if it's decorative for the spec, can help the unfamiliar reader who doesn't have all the parsing rules in mind.
Now, we can discuss whether we should optimize for the unfamiliar reader, and whether the illusion of actual meaning the trailing slash in HTML5 can be harmful.
I would note that exactly like trailing slashes, indentation doesn't mean anything for the parser in C-like languages and can be written misleadingly, yet we do systematically use it, even when no unfamiliar reader is expected.
At this point, writing a slash or not and closing all the tags is a coding style discussion.
Now, maybe someone writing almost-XHTML (closing all tags, putting trailing slashes, quoting all the attributes) should go all the way and write actual XHTML with the actual XHTML content type and benefit from the strict parser catching potential errors that can backfire and that nobody would have noticed with the HTML 5 parser.
It doesn't indicate that, though. If you write <div />, for example, the content that follows is inside of the tag. So the reader still needs to remember how HTML works, because the slash does nothing.