1. The first method is to copy an Illustrator object with a complete path, and paste it directly to a solid layer in After Effects. A path that you copied from Illustrator will create a mask on the solid layer. You can copy multiple objects from Illustrator and paste them all in a solid layer at the same time, but your solid will end up having several masks, based on the number of objects you copied and pasted.
2. The second method is to import the Illustrator file to After Effects, as you would do with Photoshop. However, there is a trick to prepare your Illustrator file in order to have the file in the proper layered format. This way, you can easily arrange the layers as a composition in After Effects.