The website in question uses PHP query strings (http://screenreader.net/index.php?colourscheme=...) to determine what stylesheet to use. That is probably done with a bit of code like this:
<link rel="stylesheet" type="text/css" href="<?php echo $_GET["colourscheme"].".css"; ?>" />
That is one solution you can use, but as you can see it's not the easiest to understand and frankly it doesn't look good in your code.
That'll create a link you can click which will change the stylesheet. If you want to get really smart, you can save the stylesheet the user selects in a cookie using the same framework and reselect it next time they visit.
The solution above may cause your page to break and display the stylesheet URL because of the use of the