(For what it's worth, in that YouTube video, they've used editing software to crop out the menu bar completely. There isn't and hasn't ever been a way to do that in the app. But there is something close!)
What a funny coincidence: I just spent the last hour making this feature more visible and more useful.
Today (in Synthesia 10.4) it's a little hard to find:
1. Hold your Shift key while launching Synthesia to open the configuration window.
2. Find the "Gameplay.PinMenuDrawerOpen" option in the list.
3. Remove the check from the Value box.
Once you close the window and start Synthesia normally, the top bar should be mostly tucked away whenever the mouse isn't near the top of the screen.
Starting in 10.5, that option will be right on the Settings --> Gameplay screen, so it'll be a lot easier to change (and available on the tablet versions of Synthesia for the first time). And, instead of it relying on your mouse's vertical position, it's going to behave much more familiarly like every video player: any mouse move or click will slide it out for a couple seconds (and it will remain out whenever the song is paused).
EDIT: And 40,000 posts here is plenty of information about Synthesia online!