Illustrator file will not rasterize in After Effects

You are looking at actual pixels in your video magnified 400%. Everything is normal. Video is pixels.

Open the Illustrator file in Illustrator, turn on Pixel Preview using the View menu, magnify the display using the same lower-left corner option to 400% and you will see the same thing. There is no error. 

Continuously Rasterizing vector layers, text, and shape layers allow you to scale them all you want, but AE makes video, and video is made of pixels, and alignment to the pixel grid is very critical for horizontal and vertical edges, so, just like in Photoshop, you have a tool that lets you magnify the pixels so you can see how the edges line up.  Having this option in After Effects is a good thing but I can see that there are several problems with your original Illustrator file because the artwork was not created with snap to pixel turned on and the weight of the line of + + + was not considered, and even the stroke around the red letters may not be an even number of pixels (points) thick, which means the graphic will never be as clear as it could be in a video. It might be clear enough, but if it were my project I would open up AI and adjust the position, thickness, and weight of almost every element in the graphic. Here are the obvious problems with the illustration:

99% probability - user error. 

Magnification Factor in the Lower Left Corner of the Comp Panel shows you pixels, not vectors.

Any raster effect on a vector layer will prevent CR from working.

Embed a screenshot with the modified properties of the layer giving you problems (select - press uu) and the source file selected in the Comp Panel so we can see it's properties and maybe we can help.