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What is the source of your type? Did you create it in Illustrator? Only true vector artwork (like that created in Illustrator) and text layers within AE are resolution independent.
If you've created the text in Photoshop, you can convert the layers to text layers in After Effects to re-enable resolution independence.
I created each letter (A is the example) in Illustrator with about 65 layers. Imported them to After Effects. They are "vector art" types in after effects.
Can't see the video due to the music copyright restrictions, but I would suspect you are using something that causes rasterization even in AI. Anything from clipping masks to pixel-based effects like drop shadows or Gaussian blur. Anyway, probably perfectly normal and correct the way AE handles it. So unless you change the setup in AI, it will stay this way...