After editing, where do photos reside?
The edited photos don't exist. The edit instructions you used (for example, exposure +0.25, contrast 20, highlights -20) are stored in the Lightroom database and so Lightroom can display your photos with these edits any time you want.
Do I need to do anything with them?
No, unless you want to do something with them.
Want to export edited photos back to my external hard drive. Can I do that?
Yes, you would use File->Export. But most people don't do this as a regular part of their workflow. Most people export only when you have a need for the photo outside of Lightroom (i.e. e-mail, web, transfer to other software), and then delete the exported photo whenever the need for the exported photo is gone, knowing that LR can always reproduce an exact copy whenever you want. I have zero exported photos sitting around on my hard disk. I just don't keep the exports. Your needs may vary.