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I can't figure this out. For a number of search terms that I compete for, there are competitors that rank below me, but their pages are featured in a rich snippet.  I wanted to see what kind of structured data these sites are providing, thinking maybe there's something I can learn.  But when I run these URLs through Google's Structured Data Testing Tool, it tells me these pages contain no structured data! So how is it that Google think's my page is more relevant (I rank higher) and I have structured data, but Google chooses to feature a different page?  Does anyone have ideas on how I can snag these rich snippets for myself?

I found a few websites that are displaying images from my site.  Curiously, they have also linked these images to the source image file on my website.  I found these because Moz showed these sites in my list of backlinks.  I assume that means it impacts my Domain Authority.  But does this actually help me in search engines since it's not linking to a page on my site?