Look at the parts for your Table-Mate. There are only five: the table top; the legs, which are bent at 90 degree angles; the U tube, which is shaped like a horseshoe with curved corners; and the support rod, which is a similar shape but is thinner and has more angular corners.
Find the two large holes in the U tube. They are on the inside of the curve. Place one end of the support rod in each of these holes. Simply slide them in and they should stay.
Place the tabletop in front of you with the warning label side down. You will see two clips of the same size next to one of the longer edges. Slip the bottom of the U tube's horseshoe into these clips. You'll hear a snap when you've done it correctly.
Turn the tabletop so that the side with the clips is furthest away from you. Pull the support rod forward and down. Note that you're pulling it toward two sets of hooks attached to the tabletop. Slide the rod into these hooks.
Squeeze one end of the U tube and slide a leg into it, making sure the opposite end of the leg is pointing towards you. Stop squeezing when the spring clip on the end of the U tube fits through the holes in the leg to attach it firmly. Repeat this step for the opposite leg.
Flip the Table-Mate right side up and turn it so that the legs are furthest away from you. Now you are free to use it any way you wish.