Imagine you're in your living room editing a photo you have on your Dropbox account using your favorite photo-editing app on your tablet, and your Roomba unplugs the cable linking your router to the Internet. Imagine your relief when you learn that your app supports syncing via a new API that Dropbox released today, so your work is preserved offline and automatically synced once your connection is back up. Dropbox, in an apparent effort to compete with Google Drive in the app support arena, has released a new sync API so your favorite app developers can address scenarios such as this.
Previously, app developers had to write the code themselves to download your files from Dropbox, edit your files while they reside on your device, and upload them back to your cloud storage (not to mention deleting your offline copy when you're finished). They also had to figure out for themselves to track the progress you've made using their apps to make changes to your Dropbox files. With this new API, Dropbox is volunteering to do the heavy lifting with these tasks.
One potential benefit of this API is that it could result in smaller app sizes (one app developer told Dropbox that the new API had cut his code for handling Dropbox integration in half). Another potential benefit is that you might not have to keep as many of your photos, videos, or documents offline if your favorite apps work with your files in the cloud just as seamlessly as if they were on your SD card (an important issue, with the recent trend towards non-expandable device storage).
Since Dropbox now provides a standardized way for developers to sync to your cloud storage (as opposed to the myriad of custom solutions they've had to implement up until now), you might see more mobile office suites, backup solutions, photo and video editing apps, etc. offering Dropbox integration.