According to the PlayStation Blog, the PS4 Pro's Media Player app has been updated to support 4K video, meaning you can use it to play just about any 4K video file from a USB stick or your home media server.
Playing your own 4K video files on PS4 Pro is simple. If you have a media server set up, it will show up automatically on the console's Media Player app. If you want to play your files over USB, just plug in any USB drive with 4K content on it. Keep in mind that any drives you've formatted to be used as PS4 external storage will not work; those are only for games and apps.
The Media Player app has also gained the ability to play 4K VR videos for folks with PlayStation VR headsets. While PSVR doesn't output at 4K, Sony says you'll enjoy improved image quality over standard HD VR videos.
This update gives the $399 PS4 Pro some extra value as a 4K entertainment machine, particularly for folks who have lots of 4K content saved locally. Unfortunately, the Pro still doesn't play 4K Blu-ray discs — you'll need Microsoft's $299 Xbox One S for that.