It sounds like you’re using Smart Sync, and the files downloading were online-only. If this is the case, then you might be experiencing this behavior.
Generally, if you're opening an online-only folder while in an app, then it is requesting information on the files. The Dropbox desktop application sees this request and then begins to download the files for you.
Try instead navigating to the folder directly from Explorer/Finder, and then opening the file in question to avoid this in future.