Doing something neatly means doing it in a tidy or orderly way. When you make your bed neatly, you tuck in the sheets and fold back the comforter.
You can put your hair in a sloppy ponytail, or you can pull it back neatly. And while your closet might be full of messy piles of shirts and sweaters, the ones you see at a clothing store are folded neatly in piles. Neatly comes from neat, which originally meant "free from dirt," and later gained the meaning "tidy." The Latin root is nitidus, "gleaming."