It's actually not all that strange (and it actually *is* a thing in Georgia, one of the few places in the US where baton twirling is still quite popular.) I started doing it when I was 13. If you already are a skilled baton twirler, moving to a fire baton isn't a huge challenge. The trick is that once you light the baton, you have to keep it moving, only stopping it in a perfectly horizontal position in order to prevent getting burned by the rising flame. You can get the moving flame quite close to your skin without a burn, because it's going so fast: just like quickly passing your finger through a candle flame. You do have to be careful about your costume, though; no components that can melt.
You can buy a standard fire baton on
Amazon; it's a vented hollow steel shaft with very high-density canvas wrapped around and screwed into each end; the canvas doesn't burn as long as gas is present, and they essentially last forever if you take care of them; I've been using mine for over 30 years now. The dancers in the show sometimes use this type, and sometimes the more traditional Samoan version that has a thicker shaft and is hand-wrapped on the ends. Traditional male dancers deliberately build very heavy calluses on their hands and feet so that they can directly expose them to flame for short periods.
The "hole" in the stage that you see him take it out of is an air-tight steel soaker box, into which about 1/4 inch of fuel has been poured after the previous show; the baton is set into it to soak up the fuel. (Batons are always fueled with camping-stove fuel; naptha.) When the baton is placed back in the box the tight lid extinguishes the flame.
FWIW, I also own a set of Samoan twirling knives. Those are exponentially more difficult to handle than a fire baton. While the edges are dull, each one weighs about 5 lbs., and the hook on the end is nasty even without a sharp edge.