Why plants grow faster in a soil?

When someone places compost in a compost pile, decomposers such as worms, or bacteria eat and poop out the compost you through into the pile. The decomposer's poop is very fertile and has a ton of nutrients for a plant to help it grow. Compost helps plants grow because it makes the soil more nutrient rich so the plant can absorb more minerals which helps it grow.
The answer is food. Compost is natures fertilizer Compost also has the benefit of adding organic matter to the soil. Always make sure you use composted material because some green or uncomposted material can actually rob the soil of nutrients as it decomposes. Leaves are a safe uncomposted material just dont use to much.

Now my answer was based the fact you were putting the compost around existing plants but if you have a garden that is dormant and have a tiller you can till the uncomposted material in then plant the next spring.