Why do vanity journals exist?

We do not call them "vanity journals." We call them "predatory journals." They make their money from people who do not understand how journals work. Their customers do not know that anybody can set up their own fake journal website. The person who is fooled by the predatory journal might be the author, or it might be the person responsible for evaluating the author's publication record. Ego stroking is not as relevant.

They continue to exist because they are profitable.

If you’ve ever submitted a manuscript that was eventually rejected, you can understand why people will resort to this kind of publication. Publishing novel stuff is hard, and not everybody can do it.

First there are the true crackpots who see the chance to finally tell the world about their pet theory. Friends have always told them they are really smart and so the only reason why regular journals do not publish their stuff is because of some conspiracy.

More common are those people for whom this is a means to an end. One is required to show a certain level of publishing activity to maintain a job or a salary level, and this is a way to maintain the appearance of productivity without skipping too many fishing weekends with your buddies.

Some do it to keep up with the Jones. Well after all I should have the same perks as my colleagues who manage to publish crap only because they know the editor, or their collaborators do all the work etc, but at least I publish my own ideas by myself...

Validation does not come from doing this yourself, it comes when your work is validated by other neutral third parties.

Of course there are also predatory conferences. Quoting from this Vancouver Sun article:

The debate over such publications often revolves around which academics are genuinely duped by the deceptive journals and conferences, and which academics take advantage of them to advance their careers.

As I pointed out in a comment this question can be answered from a lot of angles.

On one level predatory journals exist because there's a demand for them. Where there is demand, someone will supply it, especially since it's profitable.

One another level, people publish there for various reasons. Some people are genuine victims, but there's increasing evidence that others are publishing there in spite of knowing the publisher is predatory.

If your question is why publish there instead of set up your own predatory journal, it's easy to think of some reasons:

• It's not as easy as you think to set up a predatory journal. Sure it's easier than setting up a real journal, but it's still not trivial. For example, how would you go about setting one up?
• It's psychologically easier. You know you're doing something that's not quite right, but you're doing it anyway because there's an illusion that it is OK. In the same way many people are OK with downloading pirated papers off SciHub without checking their local laws even though they probably know they may be doing something illegal.