Importing Illustrator Doc Issues

You shared a PDF that was only 307 X 172 points/pixels. 

Your AI artboard should be the same size as the comp frame, snap to pixel should be turned on in the View Menu in AI, you need to avoid lines that are less than 2 points (pixels) thick and thin lines must be aligned to the pixel grid, and you should always check a Pixel Preview in AI at 400% or more to make sure your artwork is lined up on the pixel grid. Then you can import as a comp, retaining layer sizes and AE will generate a composition that you can animate.

Each element you want to animate should be on a separate layer, you should not apply any Raster effects like Drop Shadow in AI, and you should watch your colors.

Even at this incredibly small size, the artwork does not have even dimensions, and the elements are not perfectly aligned, but I can still scale up that PDF to 1500% and still get sharp edges.  

There are a couple of good articles on preparing images for video. You should check them out. You should also spend some time with the User Guide so you have a better understanding of the UI. 

One more thing. It is a good idea to save your illustration as an AI file with PDF Compatibility checked. PDF's can get funky pretty quickly.