Why Hyperinflation Happens
Hyperinflation happens when a government persistently finances its deficits by printing money and the public loses faith in the currency, so money growth, prices, and expectations spiral upward together; it ends only with a credible fiscal and monetary regime change.
Hyperinflation is different in kind, not just degree, from ordinary high inflation. It is almost always a fiscal story wearing a monetary mask.
The mechanism
A government runs deficits it cannot fund by taxes or borrowing, so it has the central bank print money to pay the bills. Prices rise; the real value of the money raised falls; so it prints even faster. Meanwhile people, expecting their cash to keep losing value, spend it as fast as possible, which makes money circulate faster and pushes prices up further. Credibility collapses and the spiral becomes self-reinforcing.
How it ends
Sargent's study of the 1920s hyperinflations (Austria, Hungary, Poland, Germany) found they ended abruptly, not gradually, when there was a credible regime change: the budget was brought under control and the central bank was made independent of the treasury. Stopping hyperinflation is about fixing the fiscal root and restoring belief, not fine-tuning the money supply.
Real cases
Weimar Germany (1923), Zimbabwe (2008), Venezuela, and Argentina (1989, ~5,000%) all fit the pattern. See the high-debt monetization scenario for the milder, more common version of the same force.