The game behind it
This is the clairvoyance game from The Mathematics of Poker. You hold either the nuts or air and you know which; the caller holds a bluff catcher, a hand that beats your air and loses to your nuts. The caller's range is capped: nothing in it beats your best hands. You bet every nut hand and add bluffs so the caller cannot profit from calling or folding, and the caller calls just often enough that your bluffs break even.
Why the biggest bet wins most
With a bet of b pots, you bluff b ÷ (1 + b) times per value bet and the caller calls 1 ÷ (1 + b) of the time. Your share of the pot is v(1 + 2b) ÷ (1 + b), where v is your share of nuts, and it rises with every increase in b. A bigger bet lets you bluff more, and each bluff turns a hand that would lose at showdown into one that sometimes wins the pot. The caller can do nothing about it, because a capped range has no hand that wins against the nuts.
A worked example
The page opens on 150 behind in a pot of 100, with the nuts 40% of the time. Checking wins the pot only with the nuts: 40%. Shoving 150 lets you bluff 0.6 times per value bet, so 37.5% of your shoves are bluffs; the caller must call 40% of the time, and you take 64% of the pot. Over two streets, half-pot bets on the turn and the river get the same stacks in. The poker math page has the bluff and defense formulas, and the pot odds calculator gives the caller's side.
Real ranges are rarely this clean. When the caller can hold the nuts too, or your value hands are thin, smaller bets do better.