A wider aperture makes blur circles larger for objects that are not at the exact focus distance. When the lens is focused at one distance, only that plane is rendered as points on the sensor. Points in front of or behind that plane are projected as small discs instead of points. The diameter of those discs depends on how wide the lens opening is: a larger aperture lets rays enter from a wider range of angles, so those out-of-focus discs become larger. Larger blur discs are easier to see, which is why the background appears more blurred.

A smaller aperture restricts the light rays to a narrower cone, shrinking those blur discs and increasing the range that looks acceptably sharp. That is why stopping down increases depth of field.

In practical terms, aperture does not change what is perfectly focused; it changes how large the out-of-focus blur becomes, and therefore how much of the scene appears sharp to our eyes.