There are multiple things that I know why Pi
1: A full operating system
The Raspberry Pi runs a full operating system with the most common one being Debian linux (Raspbain). This means that you can use a wide range of programming languages from Python, Java (Debian Soft Float) , C, Node.Js, PHP ect! It also allows a GUI interface as default can be used as even a basic linux computer for web browsing and document making. The Raspberry Pi now has the ability to run ANY ARM based operating system, not just the ones using the old arm chip.
For me one of the biggest points is the price, being a student it cost me as much as a computer game does or even some textbooks I have to buy.
One of the most important things about the Raspberry Pi is the community, there is official accounts on Google +, Twitter and Facebook along with their own website which blogs about projects, news and has a forum for you to ask questions and show off your projects. There is also sites like this one setup for support, an IRC channel and more!
4: All in one unit!
If I wanted to run a website server on an Arduino I would first need to buy an Arduino and Ethernet shield which would cost about the same as a Raspberry Pi if not more. And then I'm also restricted to just HTTP and not stuff like PHP. If I want to display TV output on an arduino I need to buy or make converters and the same to interface it with an SD Card. From where the Pi is a full blown device with OS it allows for this just to be done without any extras needed from the start.
5: Power Consumption
Compared to the arduino the Pi still uses slightly more power consumption but a lot less than a computer or laptop! The Model B uses about 5W of power.
6: Space is your limit!
Another thing about the Pi is how many things you can do with it from being able to control RC tanks from it, send it to space and take a picture, use it as a home media center, host your very first website on it, play minecraft on it.
7: Constant improvement
The Raspberry Pi is being constantly improved, it now comes with 1GB of ram, the ability to overclock to 1Ghz without voiding the warranty, currently converting from X11 to Weston / Wayland, Special Camera Module! The team has not stopped working on it since the launch.
And that is my view of the Pi, for the record I have done quite a few things with it now (http://www.raspberrypi.org/archives/1298 & http://www.raspberrypi.org/archives/1677). I also now own 4 of the devices and don't regret waking up at 5:55 AM on the 29th of February 2012.