When Apple released iOS 6 and removed the native YouTube app, Google released its own YouTube app. However, unlike the native app in iOS 5 and earlier versions, Google's version doesn't allow you to play the audio from videos in the background. See also: How to download YouTube videos to your iPad
The same is true in iOS 7, but there is a way to listen to YouTube while using another app - or in standby mode - on your iPhone,iPad or iPod touch running iOS 7.
Here's how to do it.
Step 1. Launch Safari, the default browser on your iPad, iPhone or iPod touch. (You can also launch another browser, such as Google Chrome)
Step 2. Search for YouTube in the address bar / search box, or simply type youtube.com.
Step 3. Tap the small magnifying glass icon and then search for a video that you want to listen to.
Step 3. Tap on a video from the list of results and let it begin to play.
Step 4. Press your device's home button to return to the home screen. The music will stop. But, swipe up from the bottom to display Control Center. Tap the play button to continue playing the audio from the YouTube video.
Step 5. You can now launch another app, such as Messages or Mail and keep listening while using that app. Some games (those which allow you to keep listening to your own music) will still be usable without cutting off the audio from the YouTube video.
If you want to browse the web, you'll need to use a different web browser from the one which is playing the YouTube video, since switching tabs will cause playback to stop. If you don't have another browser installed, try Google Chrome.
Step 6. You can listen to YouTube with the screen off, too. Press the sleep/wake button to turn your device off and the audio from the YouTube video will stop. Now press it again and tap the play button to restart the audio. Now when you turn off your iPhone or iPad, the audio will keep playing.