Assuming you are using the iPhone to do the transfer the problem is probably selecting the photos in DropBox. I've done some checking and there does not appear to be a way to select all the photos at once. This means that you have to tap each one individually - a lot of work for 1,000 items! Once they are selected though, you tap Export at the bottom of the screen and choose Save Image. They will all then appear in the Photos app.
If you have access to a Mac computer it's a lot easier - as long as you have iCloud sync turned on on both the computer and iPhone. Make sure you can see the Photos app icon (in the Dock or in the Applications folder) then open the DropBox folder where the photos are and select all (Command A). Finally drag them to the Photos app icon and the app should open for you to confirm the import. Then just wait a while and the photos should be uploaded to iCloud and downloaded to the iPhone. This up- and down-loading may take a while.