A lot of a vegetable's vitamins and fiber are located in the skin...so who decided that we had to peel the skin off of our carrots and potatoes in order for it to be edible? Its completely natural to eat a carrot right out of the garden with skin on. The dirt won't hurt you...just scrub the veggie under cold water with a scrub brush. I also leave the skin on potatoes when making mashed potatoes...it saves so much time and adds more nutrition and flavor:)
So unless you are entertaining guests who would be totally offended by the site of a carrot in its natural state...leave the skin on.
...it doesn't look so bad, does it?