Soaking in warm water relaxes tense muscles, warms the body and helps many people unwind. Part of the effect comes from the heat and the buoyancy of the water, which would be true of any warm bath.
Onsen add minerals, and each spring is different. Sulfur springs smell of eggs and are traditionally loved for the skin; chloride springs are said to keep you warm after you leave; simple springs are gentle and easy for first-timers. In Japan each spring has traditional effects listed on a sign, but these are customary claims, not medical advice.
Stay safe. Hot water raises your heart rate, so keep each soak short, take breaks, and drink water. Do not bathe after heavy drinking or when you feel unwell. If you have heart disease, high blood pressure or are pregnant, talk to a doctor first.
Enjoy it with good manners: wash and rinse before entering, keep your towel out of the water, and ask about tattoos before you pay. Source-flowing water is the freshest way to experience a spring, which is why this site lists it.