Any Widget-able app would act like a folder does now. When tapped it slides open and shows some details in a square area. Just like a folder, tapping outside the area closes it. Moving and deleting would work exactly like a normal app icon. No new UI to learn at all.
The widget area would be a normal UIView, so it could scroll or do pages--anything an app can do, only limited to a 320x320 point square.
The app could add an "Open" button to itself that opened the app full screen as it does now. That leaves the UI entirely up to the app: Weather could be as it is in the picture, iMessage could read or send from there, Facebook could post, Instagram could take pictures, you could play any number of games in a nice square area, anything.
Updated, Jan 2, 2013:
UI Widgets could be triggered by other parts in the iPhone system as well, and direct to different "widgeted" parts of the app. For example an email app could have a widgeted "Reply" function for its new mail notification. A quick cut-and-paste:
The UI could be much better--replace the notification message with the Cancel/Send stuff at the least.
Each app could define its own widget bits, and what a notification action triggered. Facebook could allow replying to a post or message, SMS could reply, some games like chess could even let you make your next move.
The same function could work right on the lock screen too.