It's a mesh helmet cover. Their primary purpose was to prevent or lessen reflections from metal helmets.

They also functioned similarly to the netting used to create gilly suits, allowing the soldier to customize their helmet camoflague by sticking local brush into the netting. As a big round helmet is a very unnatural sight, it sticks out, adding a couple branches and leaves would break up the silhouette. This British soldier has stuck frayed pieces of burlap cloth into his helmet cover for the same effect.

Modern militaries don't really use them anymore, for a couple of reasons. The kevlar helmet (non-metallic, so less of a glare problem) has been in use since the 1980s, with the first combat use for US forced being Grenada in 1983. Modern (relatively) camouflage is also more efficient than the old olive patterns, so there's less of a need for the mesh cover. I won't get into more recent digital camo to avoid running afoul of the 20 year rule, though shout out to CADPAT, first developed in 1995 by our northern neighbors.

There was no official nets for US armed forces until 1944, late in the war.

The marines were the first to get a cloth camo cover in late 1943.

Prior to the M1-1944 net being issued, US forces "Improvised" with canadian or british nets, already used with the Doughboy helmet that was used by US forces early in the war.

As for use, it was simply to add bits of foliage.
You need a rather close weave net to reduce reflection/glare.
Regular olive drab paint, or even campfire charcoal would reduce glare better.

Any type of cloth cover will also reduce impact noise if moving around structures and ducking under beams, etc.

This image from 1943 will show you that a large variety of field-improvised nets were in use.
http://i.imgur.com/59A6U.jpg

This thread has a lot of knowledgable guys commenting on different types of nets.
http://www.usmilitariaforum.com/forums/index.php?showtopic=1894&hl=helmet,and,net

Another thread with many images of styles and variations.
http://www.usmilitariaforum.com/forums/index.php?/topic/181999-camocanadian-style/