The short answer
Kelly sizes a stake as a fraction of your bankroll, scaled to how large your edge is. A bigger edge stakes more, a thin edge stakes little, and no edge stakes nothing.
The formula for a simple two outcome bet is f = (bp - q) / b, where f is the fraction of the bankroll to stake, b is the decimal odds minus one, p is your estimated probability of winning, and q is one minus p.
It was published by John Kelly at Bell Labs in 1956, originally about signal transmission rather than betting, and it maximises the long run growth rate of a bankroll rather than the expected profit on any single bet. Those are different objectives and the distinction is the whole point.
A worked example
Suppose you are offered 2.50 on something you estimate has a 45 per cent chance.
Here b is 1.50, p is 0.45 and q is 0.55. So f = (1.50 x 0.45 - 0.55) / 1.50, which is (0.675 - 0.55) / 1.50, which is 0.0833.
Kelly says stake 8.33 per cent of your bankroll. On a bankroll of 1,000 units that is 83 units, which strikes most people as a great deal more than it feels comfortable to put on a bet that loses more often than it wins.
That discomfort is informative rather than irrational, for the reason in the next section.
Why most people use a fraction of it
Full Kelly assumes your probability estimate is exactly correct. It is not. It is an estimate, and the formula is unusually punishing about errors in the direction of overconfidence.
If you overestimate your edge, full Kelly does not merely stake slightly too much. It can stake enough to turn a genuinely positive edge into long run bankroll decline, because the losses compound faster than the wins recover.
So the common practice is fractional Kelly. Half Kelly, or a quarter, staking half or a quarter of what the formula recommends. Half Kelly gives up a modest amount of theoretical growth in exchange for a large reduction in the swings and a large margin for being wrong about your own edge.
The Kelly staking calculator on this site returns both the full figure and a half Kelly line for this reason.
What the Kelly criterion assumes
It assumes you know your probability. This is the assumption that fails in practice, and everything else is secondary to it.
It assumes bets are settled one at a time, so the bankroll is updated before the next stake. Real betting involves simultaneous positions, which changes the correct sizing.
It assumes you can stake any fraction. Bookmaker minimums, maximums and a restricted account all say otherwise.
It assumes the only objective is long run growth rate. If a run of losses would end your involvement, either financially or because you could not tolerate it, then growth rate is not actually your objective and Kelly is answering a question you did not ask.
How to use it in practice
Use it as a ceiling rather than an instruction. Kelly tells you the point past which staking more is counterproductive even when your edge is real, which is useful information even if you stake well below it.
Be honest about the probability you feed it. A generous estimate produces a confident number that is wrong in the most expensive direction.
Recalculate against your current bankroll rather than your starting one, since the whole method is proportional.
None of this is a projection of results. Sizing correctly does not make a bet win, and no page here will tell you what any approach might return.