How do I transfer my After Effects file to another computer, without losing the images?

I'm not sure why you feel it's necessary to move the project to a different machine.  You can certainly render out the completed AE project in a media container & codec that the other machine can read.  Then it's a simple matter of transferring a single media file.

It's too bad you have to use iMovie.

Unlike Photoshop, but like Premiere pro or any other non linear editor, your project files contain only references to your footage. The original images or video are never effected with you edit or change them in After Effects. If you move the original files, or delete them, then After Effects cannot find them any more.

If you want to share a project either Collect your files, or copy the exact structure of your folders and copy all of your source materials and your AEP file and send the whole thing to a portable drive to move the project to a new computer.

This is the most basic of basic AE skill you should have. Please study up. You cannot be successful producing video if you don't learn how the process works.

Collect Files:

Adobe After Effects CS4 * Collect files in one location

This is a CS4 document - sorry, I've spent about 10  minutes trying to find a more recent version and Adobe's search engine sends me on a wild goose chase each time! - but it's essentially unchanged in any subsequent AE version.  Note that in CC and CC2014 the Collect Files command is in the sub-menu  File/Dependencies.

Edit:  After a bit more scrounging I tracked down a more recent reference:

After Effects Help | Basics of rendering and exporting

Basics. Really. File --> Replace or File --> Collect before transferring your data. and considering the mess you made, you might want to learn the fine art of creating and working with a dedicated project folder and copying everything you need to it in the first place.

Mylenium