I don't believe you can export directly from AE to Final Cut Pro, however, you CAN export from AE to PrPro, which can export to a Final Cut Pro XML.
To do this:
1. In AE, go to File>Export>Adobe Premiere Pro Project
2. Save the project where you'd like it.
3. Open the new project in PrPro.
4. In PrPro, go to File>Export>Final Cut XML
5. Save it where you'd like.
6. Done! Open or Import that into Final Cut Pro.
If you are talking about rendering then just use any production codec that FCP supports. You have access to them all. Use the Render Cue or create a custom preset in the Adobe Media Encoder.
If you want to do something like dynamic link with FCP, that does not exist. You can save a PPro project as a FCP XML but any PPro effects or PPro settings unique to PPro will not come across. If you turn an After Effects project into a FCP project by going through PPro as was suggested then you will NOT get any of the effects you applied to your composition or any of the animation or anything but the footage In and Out points, opacity, rotation and position because that is all that will be moved into PPro because After Effects plug-ins are not compatible with PPro. If there is a dynamic link to an AE project in your PPro project that will not move to FCP at all. You'll get nothing.
So the long and short of it is if you want the work that you have done in AE to show up in FCP you must render the project. I use ProRez 444 10 bit for any work I do on FCP. You can make a custom Output Module preset to help speed up the workflow.
If you are new to AE please read this on rendering: After Effects Help | Basics of rendering and exporting
If you are really new please go through these three tutorials: Basic Workflow
From After Effects, add your comp to the AE Render Queue. In Output Module, set it to render a Quicktime Movie in ProRes 422.