There are a vast number of reasons why you should keep your browser(s) updated. Let's touch on a few of them, so that you, the regular web user, can understand why this is one of the most important things you can do!
1. Developers and Other Web Users Won't Hate You.
For those of you who enjoy being well-liked and well-respected, this is important. Nothing is more important on the Web than frequent updates; nothing kills frequent updates faster than disgruntled developers. NEWS FLASH: we hate having to build workarounds for old tech; as time progresses, newer features are released that make our jobs easier and produce more appealing and intuitive results for users. Stop holding us back. On the same token, other users who have already upgraded to the most recent browser versions are also expecting us to produce content that uses the new tech. Sticking to your old-ass browser not only makes developers hate you, but ruins the User Experience for other users.
2. Security Patches
Hey, I'm a security expert. Companies pay me to ensure that whatever data you send them is secured on their servers, and that no external or unwanted access occurs -- but we security experts are only one side of the equation. On the other end of the tube is you, the user. If you don't take a proactive stance regarding security, attackers can compromise your data with crafted client-side exploits, making all of our work useless.
3. Progressive Enhancements
In addition to security patches, newer versions of browsers often include features that make your Web experience better. Browser developers, like all developers, tend to take user feedback into consideration when designing and implementing new releases. If you're not running the latest versions, not only are you missing out on better website technologies, but you're also missing out on better software technology that makes your browsing more convienent.