Why does a smaller aperture (higher f-number) give more depth of field?

A narrower aperture admits light through a narrower cone converging toward the focus plane, so as a point moves away from true focus, the resulting blur disc grows more slowly than it would through a wider cone — the same acceptable-blur threshold (circle of confusion) is reached further from the focus plane in both directions, which is exactly what a larger depth of field means.

The Cone-of-Light Picture

Picture the light from a single point converging to a sharp focus and then diverging again on the other side of that plane — the wider the aperture, the wider the angle of that cone, and the faster the resulting disc grows as you move away from the point where the cone is narrowest. A narrow aperture flattens that cone's angle, so the disc stays under the acceptable-blur threshold across a wider range of distances. This is the same geometric reasoning behind hyperfocal distance shrinking as aperture narrows: N sits in the denominator of H = f²/(N×c) + f specifically because of this cone-angle effect.

Where This Stops Helping

Diffraction sets a real practical limit: past a certain aperture, typically somewhere in the f/11-f/16 range depending on sensor pixel density, the sharpness lost to diffraction spreading across the whole frame starts outweighing the sharpness gained from extended depth of field. Stopping down further than that point can make an image look softer overall even though its theoretical depth of field keeps growing — the depth-of-field benefit and the diffraction cost move in opposite directions from the same physical cause, a narrower aperture.

See exactly how much depth of field a specific aperture change buys you in the DoF & Hyperfocal Calculator.