just pull them.  Alternatively, you can use a digging fork to dig them up. 
 
You know carrots are ready for harvest by pulling one or two and seeing if they are big enough.  I also dig around the top of the carrot and see how large the carrot diameter has grown. 

After you harvest, try to replant the area with something else like lettuce or beans.  I don't recommend planting more carrots in the same spot more than once a season if possible.  If you must use the same area, be sure to aerate the soil and add manure or other compost.