There isn't an "Exit" or "Close" feature in the application. You can just back out or hit the home button on your device to leave the app. Android doesn't really "close" applications so even if an exit button is included, it functions the same as backing out of the app with the "back" button after entering it. Android shuts down applications as it needs to in the background.
When backgrounded, 1Weather will use minimal battery and data. You can set a long update period, such as 12 hours, to minimize any data usage when backgrounded then manually update if you wish.