The anti-backlash nuts are an adjustable pair of nuts that transfer the rotation of the leadscrew into the linear motion of the axes. The two screws allow you to adjust the tension between the two nuts to eliminate backlash (the wobble that results from wear in the nut or leadscrew).
In this case the nuts are made from 0.5 inch thick delrin (bought off e-bay), a plastic that is often used for moving parts. I have made the smaller nut wider than in the plans to allow for upgrading to a bigger leadscrew. The leadscrew in the picture is M8 allthreaded rod. It is not the most efficient leadscrew by anymeans but it is very cheap and easily available. It has been difficult getting lengths longer than 1 metre in the shops and mail ordering risks the screw getting bent in the post but I have eventually found a local supplier.