You can go about this by clicking the “Quick Action” button to the right of the file you want to export, select “More”, then “Export”, and click “Save to Device” to pick your external SD card. (If you don’t see your external SD card listed there, choose the menu within the “Save to” screen, followed by “Settings” and then “Display advanced devices”.)
For those who don’t know, Dropbox is a cloud storage app similar to iCloud, OneDrive etc. in which you can store your files and access them from your PC or mobile device. There’s also a 2 GB free space for you to start with, once you register. When it’s full you will then have to subscribe to continue storing your files and documents.
You can head over to Play Store and download the app as it’s being rolled out. The current version also includes support for the upcoming Android L.