In simplest terms, compost improves the soil so that the plants growing in that soil will be healthier and more productive. Whether it is spread in a layer on the surface of the soil or is worked down into the soil by hand or with a digging tool or tiller, it serves primarily as a soil amendment. Compost is not a fertilizer, although it adds some nutrients to the soil. It is not a pH moderator like limestone, though it can (alter)the pH of the soil. Basically, it is humus, a component that "conditions" soil, improving its quality, texture, and structure so that it provides a more effective environment for plant roots that grow in it. While it directly improves the soil, it is, of course, indirectly improving plant quality of life.
A compost fork is different from a digging fork in that it usually four or more tines. It is a handy tool for moving compost around the property.