Pulling and stretching the candy is intrinsic to the texture of taffy, but it can be dangerous. Be very careful not to burn yourself when pulling taffy. Do not handle the candy until it is cool enough to touch, and beware that candy that feels cool on the surface can be burning hot underneath. I recommend wearing a pair of latex gloves (or two) while pulling candy, to prevent burns and overheating. If you do not wear gloves, make sure your hands are well-oiled so the sugar will not stick and burn your skin.
To begin pulling the candy, lift it from the work surface with an oiled scraper or spatula and push it into a cylinder. Stretch it between your hands. It will be very soft and will sag in the middle.