The style of cut is called julienne. True, a mandoline can make a julienne cut, as well as thin slices. However, you can also use a knife to achieve a julienne cut.
First, cut the carrot into manageable lengths--2 or 3 inches. Cut a piece into 1/8 inch slices lengthwise. Stack several slices on top of one another and cut lengthwise through all layers to make "matchsticks" or julienne-cut carrots. It works best to have a large slice on the bottom of your stack, not a small one or the round side of the carrot. A large slice on the bottom is more stable, so you're not as likely to julienne your fingers. Yes, cutting carrots this way is a lot of trouble, but this is the way to do it with a knife.
There are also other ways to get a similar result without a mandoline. If you have a food processor, try the large grating disk. You can also use a salad shooter or other slicer/shredder--there's even an attachment for the ubiquitous Kitchen Aid mixer. Even a plain old box grater can be used to grate carrots. Grated carrots may not be as neat and tidy as the ones done commercially, but they're still a good addition to a salad.
Oh, if all else fails, you can usually buy the shredded carrots in the produce section, near the bagged salads.
Best Answer
It's to clean the filter. With the structure of a coffee filter, rinsing and using a towel or kitchen sponge won't be able to clean the tiny crevasses of the filter, hence the bristles are there to penetrate the tiny holes.
You would really want to properly clean out your coffee filters, as oils released by coffee could eventually build up in the holes, making your next coffee result in a bitter taste.