Most of it looks like frame by frame believe it or not. You could probably do it with keyframes procedurally using shape layers and stuff, but this style lends itself to Flash and hand drawn work.
I'd start here, or just start fiddling with shape layers and offsets, but the explosions are definitely all frame by frame (most likely in Flash). I say Flash because it's the most common Adobe app amongst mograph designers that we all have access to that also has onion skinning and easy vector drawing tools. There are lots of tools for this, but the effect is achieved regardless of the tool when it comes to frame by frame.