For some geometric shape, they assign to every point in a space a number indicating the distance to that shape.
Not only that, this number is signed: negative means you're inside the shape, positive means you're outside.
This turns out to be very useful for writing creative and interesting shaders. If you know the signed distance function of a 2D shape, you know how to draw it. All you have to do is check whether a point inside or outside of it, by checking whether the signed distance function is positive or negative.
Furthermore, it turns out that knowing the distance to objects in a scene is extremely helpful when writing 3D ray-tracing renderers as well. This is because distance functions give a lower bound for the distance at which a ray will hit an object, meaning that you can safely march rays forward that distance without marching through an object. This technique is fittingly called ray-marching, or more specifically: sphere-assisted ray-marching, since it relies on the radius of the sphere hitting the nearest object.