I've gone through and read several articles about rich snippets and I'm fairly confident that 99.9% of business owners would be lost in reading those articles. My goal is to explain the importance of these rich snippets without getting too techie.
What's really important to a business owner is...how will this generate more business?
The answer to that is, rich snippets can help a business get more search traffic. If that business is successfully converting and nurturing leads, that additional search traffic would eventually become increased revenue.
To understand how adding rich snippets to your website can help with additional traffic, you'll at least have to have a basic understanding of what they are. One of the most important things we attempt with search engine optimization (SEO) is to make it so your website is AS EASY AS POSSIBLE for a search engine to understand what your website is all about. The more they understand, the more likely you are to rank and rank using some of the search results that stand out from the plain old result. Rich snippets help a website achieve that by blatantly telling the search engine what a piece of content is.
For example, you can tell a search engine specific business information by using rich snippets around a business name, the business address, the telephone number, the logo and so on. It would be like me walking around with a shirt that says, "Name=Josh" in giant letters. Those pieces of content may have already been on your website, but if they are just in html paragraphs, you aren't taking the extra step to make it really easy for a search engine to understand.
The other reason rich snippets can help you achieve more website traffic is that your search result will stand out from the rest. You probably see this when searching for restaurants or hotels when reviews show up directly below the result. These show up because the website tells a search engine, "hey...these are reviews" using rich snippets. As we start seeing more and more of these advanced and enhanced results, the more rich snippets will come into play.
That's the most simple explanation of what they are and why they are important. There will be some that are more important to some websites than others, but your SEO and/or website developer should be talking to you about how they should be implemented for you.
Here's a video from Google talking about Rich Snippets.