So then, uninstalling from a local computer does not affect the master files themselves, presumably stored in the cloud.
Correct, though there aren't any "master" files (more on that below).
what exactly is stored on each local computer, then? Just pointers to the master files?
Each computer that has Dropbox installed will have a full copy of the files in your account, assuming you don't use Selective Sync or Smart Sync, which are features that let you remove local copies of folders and files.
As long as you've uninstalled Dropbox from a computer, or unlinked a computer from your Dropbox account, you can safely delete the local copy because they're not syncing with your account anymore.
Seems wasteful to install full copies locally, especially if the idea is to conserve space on local computers.
But that's not the purpose of Dropbox. It can help save space on your local drive with features like Selective Sync and Smart Sync, but it's primary purpose is to sync everything in your local Dropbox folder with any other computer linked to your account.