Why can I only use a webbrowser when I use Wireless Leiden Internet?

The reason is that we want to prevent undesirable or illegal use of the (sponsored) internet connection.

Anybody can use the local network of Wireless Leiden, provided he/she compies with Dutch law. When you use the Internet you are using a gateway that only passes webbrowser traffic. This is a requirement of the Internetproviders to prevent abuse (such as sending spam).