Adding compost to any soil increases its population of beneficial micro-organisms. Their activity is central to efficient chemical breakdown of compounds in the soil into nutrients in a form that plants can take up. Although the decomposition process in compost has slowed down, billions of microbes remain in every handful of the finished material. They are incorporated into the soil when compost is. Some of the microbes are nitrogen fixing bacteria that convert nitrogen from the air into a form that is usable by plants. They are basically fertilizing bacteria. Other microbes manufacture antibiotics that protect plants from various diseases. (See below for more discussion).
Perhaps the most important microbial activity involves ongoing decomposition of the organic matter in the compost and of any organic matter already in the soil. Plants need their nutrients to be in a liquid or gaseous form. They can't use nitrogen that is bound up in a piece of decomposed grass blade. The nitrogen, phosphorous, potassium, calcium, boron, and all the other nutrients in organic materials that are needed by plants are converted by the bacteria, fungi, and other microbial denizens of the soil into usable gases or liquids. Microorganisms in compost added to the soil reinforce the resident microbial population, increasing the conversion activity and, therefore, the availability of nutrients to the plants growing in that soil. It follows that any soil to which compost has been added will offer increased benefits to plants.