Try verifying local files: in Galaxy, select the game in the left column, then click on the More button, and in the Manage menu select 'Verify / Repair'.

Try lowering the game resolution, and switching to Windowed or Fake Fullscreen display mode, to see if that will make a difference.

Are you shutting down all non-essential programs before starting the game, especially anything audio/video related?

One person in the original release of D:OS reported that the videos stopped working, which turned out to be an issue with the MS VC 2008 redistributable files; re-installing those got them playing again. For the Enhanced Edition, try uninstalling the 2013 redistributable package, reboot then reinstall.
C:\Program Files (x86)\GalaxyClient\Dependencies\__redist

Try opening a video in Windows Media Player, pause it, start the game from the executable, start a new game and see if you get the intro cutscene.

Try doing a clean boot and then test the game. Click Start, or hit WinKey-R, type in msconfig and hit enter; in the General tab, click Selective Startup, uncheck Load startup items (if required) and leave Load system services and Use original boot configuration options checked. Next, click on the Services tab, check the box to Hide all Microsoft services, then click the Disable All button (maybe make a note of which are currently enabled/disabled), then click OK and reboot the computer.
Run msconfig again to switch back to the normal boot configuration.

Try creating a new Windows administrator user account, switch to that account and try starting the game from there, directly from the executable.

One person with sound in the videos but no in-game sound fixed it by using a pak extractor program (topic / download) to extract the contents of the Sound and SoundBanks pak files. You could try that, as well, with the Intro.pak and Video.pak files in the '..\Divinity Original Sin Enhanced Edition\Data\Localization' folder.

After starting the program, switch to the 'PAK / LSV Tools' tab, in the 'Package path' browse to the install folder's Localization folder and select a pak file, then for the 'Extraction / source path' copy/paste the path from the path listed above, then click the 'Extract Package' button.

With the pak files extracted, you should be able to view the .bik videos using a player from the Rad Game Tools website, or some media players support the bik videos format, like PotPlayer (Korean homepage, English downloads: here or here).
Splash_Logo_Larian.bik is the video that should be played each time the game is launched, while Intro_Scene.bik is played when starting a new game, after character creation.