Tap the Dropbox icon at the bottom of the initial script-selection screen. (It's blue; it's a box — you can't miss it.)
You'll then find yourself on the Dropbox list screen. Tap the Account button, and enter your Dropbox email and password (i.e., the same email and password you would use to log onto www.dropbox.com).
Fade In Mobile does not save your password: it uses it only to obtain authorization from Dropbox to access the "Fade In" folder of your Dropbox account.
Note that your .fadein screenplay files must be stored in the "Fade In" subfolder of your Dropbox account.
Note also that Fade In Mobile uses a slightly different developer API to connect to Dropbox than, for instance, the official Dropbox client. It is possible that, for a short time, one may be temporarily unable to connect while the other can.