In this video this guy is surfing a huge wave. If a wave like this falls on top of you is the danger that you will pushed so far under water you will hit the bottom hard and cannot surface, or the physical impact of all that water coming down around you?
Also the turbulence of the water can make it feel like you are being tumbled around and ripped apart on even normal sized waves. I can see things like whiplash and dislocations happening, possibly collisions with other riders or the board.
I read a couple of books about the guys who do this. Pretty interesting. As I recall, the folk who died drowned, rather than from impact trauma. But, I could imagine it could be a combinatioon. Wave knocks you under, big wave increases time/difficulty of surfacing. worsened if big wave happens to knock the wind out of you or knock you unconscious.
One of the biggest big wave surfers is (was?) married to Gabby Reece. Just not fair that someone gets to live in Hawaii, make a living surfing AND have children with a goddess! 
Falling off your board and hitting the water at high speed can be dangerous if you happen to land at a bad angle. An acquaintance needed back surgery after falling while water skiing; I suspect big-wave surfing speeds are at least as high as that, so there’s one hazard.
Getting slammed into the sea bed is another hazard. If it’s coral and rocks then any hard contact is going to hurt, but even if it’s just a sand bar, head-first can be fatal.
Mark Foo died 20+ years ago while big-wave surfing. The prevailing theory is that his leash snagged on something underwater, and the rushing water prevented him from freeing himself before he drowned.
Have you seen Laird Hamilton? Add that rare ability to be a male model and a man’s man to the list. The guy is smart, innovative, a ground breaker always pushing the envelope, good looking and cool as a cucumber, he’s basically got it all. I’m pretty sure he pioneered tow-in surfing of large waves and using a hydrofoil. And yes, they’re still married, reportedly quite happily.
The recorded accounts I’ve listened to from surfers at Maverick, Chopes, Jaws, etc. talk about the disorientation of the turbulent water causing them to be held under for long periods, then right when they might break the surface the next wave lands on them and it starts over again, then again. Each of these areas have their own peculiarities though and, as mentioned, several factors like depth and bottom configuration can come into play.