There are a number of reasons why as users, we should use updated browsers.
In years past, browser manufacturers cared less for complying with global standards for how the languages of web pages (such as HTML, CSS and JavaScript) should be interpreted. It's difficult to use a language if it's subject to different interpretations. A difference of interpretation can yield objects not sitting on the page where they should or broken interfaces.
Newer browsers, including Internet Explorer 10 and 11, do a much better job at complying with standards. HTML 5, the latest version of the language of the web, offers new behaviors and features, that new browsers should be compatible with.
As new features get proposed and ratified, they need to be implemented by browser manufacturers.
An example is the box-shadow property, where one can specify that part of a web page have a drop shadow of some color and transparency and size and "fuzziness". Prior to this feature, web designers had to painfully make drop shadows by making graphic images containing the shadow. Browsers that have incorporated this box-shadow feature allow web designers more freedom and speed.
While web designers try to target a set of browsers that represent the bulk of the intended audience, an older browser may not render the website the same, as it was ideally intended to look and behave.
Web transactions follow a client-server model. Security is maintained by both the client (web browser) and the server (web server providing a website or software service).
Server providers must keep abreast of security issues and maintain software that's updated to prevent hacking. Browsers, too, must be kept updated.
Users should keep their browsers updated to currently-supported versions that will continue to receive security updates.
Recently a vulnerability was found in one of the security protocols for encrypting pages, SSL v3. Many websites, including Big Picture-hosted sites and government and credit card processing gateways, have disabled support for SSL v3, so a browser can only use the newer protocols such as TLS.
While this SSL v3 method was outdated and not used by modern browsers, Internet Explorer 6, a once popular web browser (Microsoft at the time had 80% browser market share, and it took 5 years to release its successor, IE7), does not support the newer, more secure protocols.
If you or your users are using IE6, you should upgrade immediately to a newer browser. Microsoft stopped support security updates to IE6 in 2010.
The dark ages of a lack of standards compliance are behind us, when designers and programmers had to implement all sorts of hacks to provide support for different browsers and the quirky bugs in IE6.
There are a number of choices available to users, although limited-audience software such as administrative tools may offer limited compatibility.
IE11 has a 376/555 HTML 5 score
IE10: 335/555
IE9: 128/555
IE8: 43/55
You can check your compatibility with HTML 5 here: http://html5test.com