Below the video you linked to is this text: "Greatest Super Stars, made in Maya and AE." So that kind of answers your question.
The geometric shape made out of dots and lines looks to have been constructed in Maya, which is a 3D app. I think that because the vertices all retain rock-solid connections even as the camera moves and rotates around within the space.
If you made the simple geometry in Maya, you could export the white lines and dots AND the camera into After Effects, and then place your text, etc. in 3D space using AE.
Or you could experiment with trying to get a similar effect that didn't include the lines connecting the dots. I think you could do that in AE alone by making small white solids, and masking them with the elipse tool. You could then make them mimic spheres by making each white circle a 3D layer, and making it "Orient Towards Camera" with the radio button that can be found in the Layer > Transform > Auto-Orient menu selection.
Distribute them in 3D space, put up your text next to each dot, and make an AE camera fly around the space. I just tried it, and it's actually pretty cool looking even without the lines.
You could do a beam effect and attach the two endpoints of the beam effect to the solids with expressions, that way even if they moved around, the line would still be attached. (Example here)
If you wanted perspective, you'd have to do some manual keyframing, but it wouldn't be too bad.
You would need an expression that referenced the perceived 2d coordinates of the 3d solids. I can't remember it off the top of my head, but it's been mentioned tons of times on this forum for making lens flares work in 3d, etc.
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