PNGS WON'T OPEN PROPERLY

Did you download a resolution appropriate for your uses? I just checked on the website you mention and the crab image. for example, is available at three resolutions: 1200 x 1200, 2500 x 2500, and 3000 x 3000.

If you are using these for a print project, the traditional rule of thumb is to work with images at 300ppi for quality output. That means the 1200 x 1200 image would be good for up to about 4" x 4", whereas the 3000 x 3000 image would be good for 10" x 10". Frankly, you could even go 50% bigger and still be good for print, as anything over 200ppi would still be invisible to the naked eye in print.

Regardless, these ARE pixel images, so the more you zoom in on them, you are going to see the pixels.

That being said, the QUALITY of the image, however, is up to the creator of it; the graininess might be a design choice or a bad compression setting. Illustrator had nothing to do with that.

Does resaving in Photoshop change it?

What are the images supposed to look like? If you reduce them in size do they look correct? If that is the case then it would be because a png file is a raster image and is resolution dependent. This means that if you enlarge the image significantly from its original size then the resolution will go down and it will begin looking fuzzy and pixelated. This is opposed to a vector image, such as native Illustrator art saved as an .ai, which can be enlarged as much as needed without a loss of quality.