If I understand your question correctly, stopping the aperture down to its narrowest ensures that light is focused as tightly as possible. If you take a photograph at a wider aperture, excess non-incident light will still make it to the sensor, and mitigate the effects of sensor dust.

To put that in more precise technical terms...with a narrow aperture, the light that strikes the sensor is from as narrow a field as possible, and as close to perpendicular as possible (90° to sensor plane). That causes dust particles to create a "sharp shadow" on the sensor. At a wider aperture, the light that strikes the sensor is from a wide field of view, and light may not always be perpendicular (could be anywhere from 90° to say 70°), since the entire lens surface plays a role in focusing light. Those off-axis rays of light cause dust particles to create a "soft shadow" on the sensor.

You could demonstrate the effect at a macro scale, if you need a visual exemplar. Hold your hand up a foot or two from your wall, and point a bright but narrow beam of light at it...say from a flashlight about 10 feet away. The shadow from your hand should be clear and sharp. Perform the same experiment again, however this time, set up several shaded lamps that emit light in a broad field in a line parallel to the wall about 10 feet away. The shadow from your hand should be soft and dim, if visible at all (except under closer scrutiny.) Narrowing your aperture is akin to using the flashlight, while widening it is akin to setting up a wider row of lights.