Turn on your iron to a "high" heat setting. Hold a cotton cloth under running water to wet it. Squeeze the cloth so it's damp, not dripping.
Smooth out the curling edge of the rug and place the damp cloth over it. Iron over the damp cloth and the rug to coax the curl out of the edge.
Flip the edge of the rug over and place the damp cloth on the opposite side. Iron over the cloth and rug. Repeat with the previous side until the curling is no longer present.
Stick double-sided adhesive rug tape underneath the corners and sides of the rug or cut a piece of double-sided adhesive mat to fit the entire underside of the rug and stick it on to keep the rug from curling up again. Stick the rug to any hard surface flooring.