Short answer - You can produce any effect you have ever seen in any video or commercial in AE if you have the right assets to start with.
Longer answer - AE is more like flying an F-18 off an aircraft carrier at night in a storm than driving a car down a country road on a summer day. Both require training, but you need a lot more training to fly a plane than drive a car. You can recreate your sample video using Illustrator and After Effects but you'll need to study up on how to use the software. It will involve layers, distortion effects and keyframes. If you are new to AE please start here: Basic AE
When you get a basic understanding of how the tools work and how to do some simple animation learn how to create layered artwork in Illustrator so that the different things you want to animate are the right size for your project and on separate layers. Then import the Illustrator file as a composition and animate the movement. When you get the movement down apply distortion effects to the layers you want to get to move like the foreground elements in this image:
I see at least 9 layers in this animation. Starting from the bottom: 9 - The purple background, 8 - The first set of blinking yellow dots (fireflies), 7 - The back layer of moving bushes, 6 - the trees, 5 = the middle row of purple bushes, 4 - The ground, 3 = The brown bushes that are not moving, 2 - The closer layer of blinking fireflies, 1 - the dark moving row of plants in the foreground. Layer 1, 5 and 7 have position and distortion animation applied. The 2 firefly layers could also be 4 layers with animated opacity so that would make a total of 11 layers in the comp.
When you get stuck come back to the forum with some specific questions. Include a screenshot of your composition with the modified properties of the layer you are having problems with revealed. You reveal modified properties by selecting a layer and pressing the U key twice. You can simply Print Screen on windows and paste the screenshot to the reply field on this forum or you can use Shift + Ctrl/Cmnd + 3 on a Mac to take a screenshot and drag it from the desktop so we can help you solve your problems.
Perhaps the wiggle shear effect on a very low setting