On using the heatmap plugin
This discussion should shed some light on how this plugin works:
the procedure in which pixel values are computed is explained in the Context Help. Its just a linear assignment, say for a buffer of 10 pixels, and decay of 0, the central pixel has a value of 1, 2nd from center pixel has 0.9 and so n upto the 10th pixel with value 0. Its then added along for point by point. So the more the number of points, the more the pixels' value. http://idvux.wordpress.com/2009/08/21/heat-maps/ has a nice explanation of various types of heatmaps (value interpolated, frequency). This tool generated a frequency heatmap. So there is not much of algo space left.
Note that the buffer is specified in pixels! So the real question is how the pixels are calculated.
In GRASS this would be specified in the region settings. If you don't mind getting used to GRASS, it offers a lot more advanced options than the Heatmap plugin does.
On styling the results
In the style tab (of the raster layer's properties) you can go to the "contrast enhancement" section an select "Stretch to MinMax". The resulting picture shouldn't be a grey box anymore but show the densities in greyscale.
If you want the results to look like on the website you posted, you'll want to select "Colormap" instead of "Greyscale". Then go to the third tab called "Colormap" and create the colors to match your data.
Tip: You'll probably want to change "Color interpolation" on the "Colormap" tab to "Linear". But give it a try.