To pick the carrots, grab them firmly by the root and wiggle them around in their hole before pulling straight up. Remove the greens immediately so they don't continue to pull nutrients from the carrot itself. Wipe off the excess dirt and if they are wet or muddy, let them dry off before refrigerating.
If the soil is particularly hard, it may be helpful to water the area where the carrots are growing before trying to pull them out. Water thoroughly so the moisture soaks down, but then wait an hour or so before starting to harvest. Otherwise, you'll find yourself in a muddy mess.