Touch Bars are great and give their users new functionalities, one of which is taking screenshots of what's currently displayed on the Bar. Unfortunately, the default shortcut for this action is COMMAND+SHIFT+6 which is also CloudApp's hotkey for entering the screen recording mode.
There are 3 ways of fixing this conflict:
1. Change the shortcut for the Touch Bar screenshots.
Go to System Preferences -> Keyboard -> Shortcuts and double tap on ⇧⌘6 and press your new combination:
2. Change the Screen Recording shortcut in CloudApp.
Open CloudApp window, click on the gear icon, open 'Preferences...' and go to Shortcuts tab.
Single tap on Record screen shortcut to change it to a different one:
3. Disable the Touch Bar screenshot feature.
Go to System Preferences -> Keyboard -> Shortcuts and uncheck Save picture of the Touchbar as a file: