Carrot seeds form on top of the green foliage of a carrot. If you let just one carrot 'go to seed', it will shoot up into lots of branches, not just floppy foliage. At the top will appear little umbrellas of tiny white flowers. These flowers hold the seeds. Cut your stem down near the carrot, and put into a paper bag. Scrunch it closed, and within a couple of weeks, the seeds should have dried out, and are then ready for planting again.
If you dig up a carrot that has a whole lot of hairs [small roots] growing down the sides, you can easily brush these off with a stiff brush. I never peel my carrots - no need - and they are much better for you as the peel is very nutritious.
By the way, you can plant carrot seeds all year round. They just grow a bit slower in winter. Leave your carrots in the ground until you need them, if possible. Pick them when you can see a big round end. Put into a plastic bag in your crisper drawer. They seem to last for over a month and they are still delicious!
This picture of a carrot was one I pulled recently: it weighed 564 grams! Perfect for eating as well.