The flashy animation occurs when you successfully complete a ring. When you get the completion ding sound and vibration, you look at your watch and you'll see a black screen and an electrified animated ring appears filling up in clockwise fashion, then the usual congratulations text and whatnot. If you complete an activity ring while doing a tracked workout, you don't get the animation, instead a pop up for a few seconds will let you know "Move Ring Completed" etc. with the respective ring color text and circle, then the screen goes back to your usual workout screen.
Do you have Goal Completions ticked on in the Activity section under Notifications in your Apple Watch app on your iPhone?