Ask yourself: How do children experience their world? Most languages have words like "grasp", that mean to understand something and to touch it.
Good schools/educators try to incorporate as many sensory channels and as many different ways to teach as possible. In my child's primary school letters and numbers are taught by having the children walk the pattern, draw them into sand, form them out of clay or pipe cleaners and only finally writing them. For something as simple as a '5'.
Now take an even younger child - how do you think they can learn about simple thinks like warm, cold, soft, hard wet, dry, sweetly scented or stinky? There is no hands-off method of understanding or explaining snow, but so easy to grasp when you feel the cold snow melt in your hand, see it glistening in the sun or caught snowflakes with your mouth.
And yes, this means getting your hands (and feet, face, clothes ...) dirty from time to time. That's why we have play clothes and water & soap for afterwards.
You ask for health impact.
In my opinion the impact of getting dirty on a child's mental wellbeing and ability to learn about his world can't be valued high enough. If you add to this that there are various studies that claim that getting dirty is good for your imune system and can reduce the risk of allergies (keyword "hygiene hypothesis") and that being active and outside (later known as "exercise" or "fitness" for adults) is good for your health too, I'm happy to do an extra load of laundry or two.
And if you need more inspiration, watch the kids in this video - playing in a forest kindergarten.