Welcome to the forum.
Boy, that title had me going, until I saw the Effect on YouTube. Color me relieved.
Now, let's take each blue dot individually. This could be done in Photoshop, or from Titler. Create a small blue dot. Import that image, if PS/PSE, into PrPro. Obviously, with Titler, this step is not necessary.
Place on Video Track 2, and use Motion>Scale, and Motion>Positon, as is requred. Then Keyframe Effect>Blur>Gaussian Blur, Keyframing the amount of Blur, and probably Motion>Scale too. Keyframe Opacity to diminish this. Add another, and repeat.
When you get to more than one on the image, at one time, just place it on Video Track 3 and so on.
You might want to also Keyframe the Levels (think that PrE only has Auto Levels, but the non-Auot might have been added in PrE 8?). Even so, for this sort of use, Auto Levels should work fine. This will make the growing dot brighter, as it expands.
One could also add some of the Effects from ProDAD's Vitascene for more "flare," etc. in each.
Good luck, and I can exhale now,