The new setting is browser-specific and opt-in, and helps users screen out potentially offensive content that they may prefer not to stumble upon (or have their family members see), even if the content is inherently not against YouTube’s Community Guidelines.
As an example, YouTube cites news videos that contain graphic violence, such as war coverage. The company is careful enough to say the filter will not be 100% perfect.
From the support page (where you can also find out how to turn on the setting, and even lock your choice on your browser with your YouTube password):
Safety Mode gives users the option to choose not to see mature content that they may find find offensive, even though it’s not against our Community Guidelines. When you opt in to Safety Mode mode, videos with mature content or that have been age restricted will not show up in video search, related videos, playlists, shows and movies. While no filter is 100% accurate, we use community flagging, hide objectionable comments and porn image detection to identify and hide inappropriate content. Safety Mode on YouTube does not remove content from the site but rather keeps it off the page for users who opt in.
The ability to opt in to Safety Mode is available in all languages YouTube currently operates in, but does not extend to YouTube Mobile at this point.