First, you also need to understand how Lightroom works. Lightroom stores your edits as instructions in the catalog. It leaves the images untouched. That's why your setup in Dropbox works. The way you've set things up, Dropbox only synchronizes the Lightroom catalog, not the actual images. Your images or image folders are not downloaded by Dropbox (nor by Lightroom). So your edits are synchronized, because the catalog folder is synchronized. And because you've got smart previews in that folder, that's all you need on the other computer. If you do need the images for some reason (for example if you want to export an image at full resolution), then you can connect the external disk with the images to that other computer.
This is how Dropbox works: Dropbox simply copies the contents of your local Dropbox folder to its servers. Then it keeps the files in sync, meaning that if you edit a file in your local Dropbox folder, that file automatically gets copied again to the server to replace the old server copy. And if a file on the server is newer than the copy on your local computer (because you use Dropbox on more than one computer and you've edited that file on the other computer), then Dropbox will copy the file from its server to your local computer to replace your local copy. That means you always have the latest copy on both your local computer and on the Dropbox server. It also means that all your Dropbox files have local copies, that's why Dropbox takes a lot of space.
You can exclude certain folders from synching to all your computers by using 'Selective Sync', but they will always be synched to at least one computer. Dropbox cannot be used to save space on your local computer by moving files to the cloud, because those files would then be unavailable to that computer. Some other cloud solutions (Apple iCloud) do allow this by mounting the server as a virtual local disk, Dropbox does not.
