The shader examples on that site are written in a C-like language called GLSL, which is the (Open/Web)GL Shader Language. When they're executed, the JavaScript WebGL code sends the shader source off to your graphics driver's OpenGL functions, which compile it down to a machine code that runs directly on your GPU.
BTW, the shader-only style of coding, while very popular and a great way of writing cool demos, is not an entirely typical use of WebGL. Normally the shaders are used to do a lot of the heavy-duty number crunching, but the design and animation of the various objects that make up your scene is done in JavaScript. (In OpenGL, you would see a similar split between GLSL and some other language, such as C or C++.)
If you want to learn more about coding WebGL, you could do worse than visit the tutorials on my site, learningwebgl.com. I frequently link to other people's demos and tutorials too, so if you don't like my examples you should easily be able to find something better :-)