YouTube is pretty strict about its stance on not allowing downloads of its videos. Even many popular browser plugins that once supported downloading of YouTube videos are now blocked from the site, so it should come as no surprise that any apps that once allowed you to do the same have been patched to remove the ability. That being said, there's still a way to do it, and here's how.
First, you'll need to download a free app called Titan Downloader. This was one of the apps that once allowed super easy downloading of YouTube videos, but has since blocked the feature -- on the surface, at least.
If you browse for a YouTube video directly from the app and attempt to download, you'll be met with a warning that, due to "copyright issues," the app doesn't allow downloading on YouTube.