More than likely, you've already got some weirdness in your life, like that guy at work who sings show tunes or the professor who stutters uncontrollably when he talks to female students or even the street preacher who condemns you to hell for going into the grocery store. That's regular, everyday, background weirdness. High Weirdness is another animal altogether.
There is no hard and fast definition of "High Weirdness", but it often includes a heavy dose of paranoia, conspiracy theory, and paranormal shit that is absent from regular weirdness. If the lights in your office flicker everytime that guy sings Gilbert and Sullivan, that's High Weirdness. If your professor stutters around women because he thinks they're mind-controlling space robots, that's High Weirdness. If the street preacher condemns you to hell for going into the grocery store which he believes is a front for the South American Nazis, that's High Weirdness.
Of course, opinions differ. What some people consider High Weirdness could be considered regular weirdness by others. Some consider UFOs and alien abductions to be High Weirdness, others feel that "The X-Files" has made the whole topic too mainstream, and others say it's not High Weirdness unless it happened in the 1950s. Some people think of any conspiracy theory as High Weirdness, but for some, it's not High enough unless it involves the Illuminati, the Freemasons, and the Orbital Mind Control Lasers. Some consider ghost stories, rains of frogs, Bigfoot, and the Loch Ness Monster to be High Weirdness; others consider them to be pop culture -- unless there are wild-eyed parapsychologists studying the phenomenon...