Key point from the presentation mentioned above by Jeremy Kendall (do watch it), that the docs don't mention until 2/3rds of the way down: to let Slim handle the routing, you need an .htaccess file:
RewriteEngine On
RewriteCond %{REQUEST_FILENAME} !-f
RewriteRule ^ index.php [QSA,L]
Et voila, the example at the start of the Slim docs will now work for you. smacks forehead
More info on the syntax and content of these .htaccess rules: