You don't. The same license works both places.
So I take my new iMac home, get onto Adobe on the internet, open my account, and then what? I need to download the software from them onto my Mac, I can't just copy the program off my PC (right?). Presumably they are going to ask me to pay (again) which I don't want to do.
WIll they give me the option of declaring that I've already paid and have a license, and can therefore download to my Mac free?
I can't just copy the program off my PC (right?).
The specific link for the Mac installer is Adobe - Lightroom : For Macintosh : Adobe Photoshop Lightroom 5.6
Download that (it's free) and it will install as a 30 day time limited trial - but with all features active.
Enter your serial number to license it permanently on your Mac.
Presumably they are going to ask me to pay (again)
No. The software download is free.