To get some better precision I do a re-vectorization of the anti-aliased bitmap using marching squares.
Any chance I could get an explanation like I was 5? 
Sure.
The marching squares reconstructs the subpixel edge (blue) from the surrounding 2x2 pixels (red):

This way the distance (orange) to the edge is much more precise; in contrast to using whole pixel positions only where the distance between two neighbouring pixels can naturally be only 1 or sqrt(2).
The only way to get a reasonable precision without subpixel-precision is to increase the resolution but that makes it extremely slow.
Another question regarding the shader that would be used to draw these distance field fonts. It would look up a texel, and it would know how far from the font it is by this lookup.... but what to do with that info? Do I draw here or not?
When fetching a texel from the distance-map a value >=192 is "inside" the character and <192 is "outside".
So in the easiest case you can just enable alpha-testing and drop all outside texels (that's compeltely un-antialiased, though).
Inside a shader you could do all sorts of color-ramping near the boundary to simulate anti-aliasing, glow, shadows, etc.
At the moment I'm not quite sure if the threshold of 192 makes much sense.
I just figured that font-characters are usually rather thin and there's probably more information outside than inside...
edit: the threshold value changed from 192 to 128 in the current version.