Long story short, the HAARP nonsense has its origins in psychology. If you want the long version, check this article. It also answers how conspiratorial thinking in general works, not just the nut jobs spreading wild stories about HAARP.

Because these delusions are based on defects of the mind, there’s no amount of facts that you can throw at this problem and expect it to be fixed that way. What wasn’t reasoned in, can’t be reasoned out of a broken mind like that. These people are in need therapy and support more than anything else.

If podcasts, are more your thing, consider listening to You’re not so smart - episode 197 conspiratorial thinking. I think the part about the history of small and large conspiracies was particularly fascinating. Grand conspiracies are impossibly difficult to pull off, because there’s always a weak link somewhere which will make the whole thing collapse sooner or later. On the other hand, small conspiracies are a reality, and there have been numerous documented cases of those happening in real life.