The toggles (the two things that expand on a pivot) on that holder are on backwards. The pivot end of the toggle should be facing the wall so as you push it into the wall it gets smaller to fit through the hole and once in the hole it expands bigger than the diameter of the hole and can not be pulled back through. [Edit by JGS: This means that if you ever remove the screws the toggles will fall into the wall cavity and be lost to you.]
Unscrew the toggles from the bolts and turn them around, but do not screw them on all the way to the holder as they are in the photo. Just screw them on far enough that are on securely then pinch the wings of the toggles together and push them into the holes in the wall. You will feel or hear them expand once they are into the wall cavity. You then pull back on the holder which pulls the toggles up against the back side of the drywall as you use a screw driver to tighten them down. Here is a video that shows how to install them. Go to the 1 minute mark to see how the bolt goes into the wall.