The target example has a black background and white circles painted on it. It also seems to have partially transparent "glows" that can overlap.
You example seems to have a white background and blue squares painted on it.
The biggest hurdle seems to me to be figuring out how to make glowing effects. You can probably figure out for yourself how to do the black background and paint circles instead of squares.
You should look into how to use colors that have "alpha" values, such as RGBA format where the "alpha" is the degree of transparency. Also, there are possibilities such as using GradientPaint or RadialGradientPaint (and the gradients can apply to alpha as well as R, G, or B).
Here's the link for
Color. Be sure and check out the info there on transparency (alpha settings) and try applying it to Paint objects.
So, paint the background black, pop on a larger circle with the desired "glow" transparent color, then put a smaller white circle over that. That might get you 80% of the way to what you want.