Backlinks are a way for websites to show a united sense of trust. The most common form of backlinks are when websites provide information of a certain type, and reference the site that may pertain to that kind of information.
Simply put, backlinks are a very vital part of SEO, because in reality, the more that other sites back link with you, the more you are trusted because you have a "backing" per say.
Another way of looking at it is that back links are a very solid way for search engines to tell if the things on your website are trustworthy or reliable. A lot of search engines also reference whether or not your website, or a website in general has backlinks, because this is a factor that they use to determine how easy your website is to find, or the ranking in position that your website has.
Another thing to keep in mind is the value that some backlinks hold. Not all backlinks are going to provide the same sense of trust as another widely known website would. Potentially dangerous, or spam based websites will not be as effective as using a reliable website that many people use.