This is what we’ve heard in the stories about WW II. The Japanese flier or soldier says it just before he executes some suicidal move.
I seem to recall reading one time that it means (or is part of a phrase that means) “May you live 10,000 years.”
True or False?
If true, then what does that actually mean? “I’m about to try to kill you, so I wish you the best?”
(Or if false, what does it really mean?)
The cheer is the traditional cheer for the Emperor. Yelling it as a battle cry is not much different than yelling “For God and Country!” It simply re-affirms the patriotic passion accompanying the act. It is found before suicide charges in American movies because American audiences want(ed) to see all those charging enemies killed.
My dictionary says:
Superscripts omitted 'cause I don’t know how to do 'em.
He was yelling to his comrades, not his intended victims. Besides, it would kinda freak you out if the enemy came at you screaming with murder in his eyes and grenades around his waist. Which the Japanese did. Often.
Damn, y’all beat me to it. And one of you had a freakin’ dictionary reference! I’m pulling answers out of my squickable parts while you people have time to look things up. Gotta learn to type faster. 
I’d say tomndebb is correct. In fact, they often explicitely said “Tennoh Heika Banzai,” literally “Long live the Emperor!” Nowadays it’s an all-purpose cheer, though a very formal one.
It was just the Japanese version of a Rebel Yell…
Yet another dictionary reference.
“Ban” from “man”, meaning ten thousand. “Zai” from “sai”, meaning years, especially age.
It’s currently mosttly used as a cheer. It’s shouted three times while raising both arms over the head, for a big event, such as a game winning home run.
Someone did it at a wedding I went to a couple weeks ago. The “BANZAI!!” thing is best done drunk, when one’s ability to synchronize cheers is shot to heck.
Japanese battle cry or a patriotic cheer, eh? How appropriate! 
And let’s not forget Banzai bonsai!, “May this tiny little tree live ten thousand years!”