When a version of software reaches end of life, it means that official support for that version stops and usually, as a flow-on effect, people who have written components/modules/plugins for that version will also stop creating/supporting their own software versions not long after.
Having said that, some people are still successfully running Joomla 1.0x websites (which reached end of life in July 2009) so just because your website is 1.5x, doesn't mean you need to rush to upgrade it immediately. But it's definitely something to keep in mind for the future. Here's why:
Problems may arise over time as underlying hardware, operating systems and software are upgraded to newer versions. For instance, most webhosts nowadays run PHP5.3, which is the underlying code that Joomla! runs on. But "old Joomla" was written for earlier PHP versions, and they won't work 100% effectively on a server running PHP5.3. Sometimes it's possible to perform minor hacks to force a site to keep functioning, but it isn't a happy relationship. So when a webhost installs a new, faster, bigger hosting server and moves all of the hosting accounts across to this new server, sometimes old websites will break and sometimes, they can't be fixed.
Browsers. Every web browser renders websites slightly differently, and newer browser versions are designed to read websites coded in a particular way. Old websites couldn't be tested in newer browsers, so sometimes a new browser version will render your old website incorrectly. It may be difficult to find a fix when your old website's code is no longer supported.
Smartphones and tablets. Websites don't magically just work on a smartphone or a tablet. Many older websites utilised Flash, which Apple don't like (and newer Android will soon also not like), so anything Flash won't even be visible. And newer, smartphone views of a website is actually a separate set of code that somebody has created, that force the content to display in a smartphone-friendly way. In a Joomla! website, this is built into the template. Template-creators have really only started doing this in the last year or 2.
Social media connectivity. Really, you're not doing yourself any favours these days if you don't at least have a Facebook Page for your business. And you want to be able to add all those fancy buttons and feeds to your existing website, to show how social-media-savvy you are. Problem is, a lot of the time nobody had written nice modules to do that, back when your website was created.
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