Here is what the actual picture looks like:
We also increased the size of the pixels (see The Mandelbrot Set Explained for an introduction to what pixels are) so that you could see them more easily. Remember, the actual size of the picture is just 32 by 32 pixels wide--on most computers, that's about the size of your thumbnail.
The black pixels represent points which are (at least approximately) "inside" the Mandelbrot set. The colored pixels represent points that are not inside the Mandelbrot set.
Basically, the computer does a bunch of calculations, and checks each time to see how big the result is. If the result ever gets bigger than two, the computer knows that the point is not in the Mandelbrot set. It then stops doing calculations for that point and assigns that point a color.
The number plotted in each colored square is the number of calculations the computer had to do to learn that the point was outside the Mandelbrot set. The number in the black squares shows the number of calculations that the computer did before deciding to give up and color the pixel black. (It is important that the computer gives up after a while, since some of the points would otherwise keep the computer calculating forever).
What this all means is that although we are pretty sure that the colored points are not in the Mandelbrot set, we don't know for sure whether the black points are in the set or not. All we know from this computer plotting is that the calculations for those points stayed small for 75 steps. We don't know, for example, if some of them might have gotten big if we had just done one more iteration.
Think of it this way--pretend that you had told the computer to quit calculating after four steps. Then everything in the picture with a number larger than four would be black. That doesn't mean that those points are in the Mandelbrot set--it means that either:
the point is in the Mandelbrot set
or
the point is not in the Mandelbrot set, but we haven't done enough calculations to find out yet