The pilots also noticed an uncommanded increase in airspeed, which they tried to check by extending panels called speed brakes, and “a pilot called out that the fasten seat belt sign had been switched on.”

An uncommanded increase in airspeed? Do they mean even by the autopilot? Very murky details here.

Finally:

It was unclear what caused the turbulence.

So, basically, they lost control of the plane and don't know what the fuck happened.

Seems to have been a few uncommanded things happening. This from another report:

Prior to the sudden drop, the plane had initially experienced “slight vibration” and an “uncommanded increase in aircraft altitude” from 37,000 feet to 37,362 feet. The autopilot pitched the plane downward to its original altitude of 37,000 feet.

Presumably it was the pitching down bit that caused passengers to hit the roof.

So there were uncommanded increases in airspeed & altitude, then auto-pilot reacts to that causing the injuries?

As you say, it's not really clear.

It happened around 7.45 am apparently. Not a usual time for thunderstorms & there was not seatbelt sign on - so it must have been 'clear-turbulence'.

Some website: Clear-air turbulence occurs in cloudless and invisible conditions .. often around the jet stream, a fast-flowing ‘river’ of air typically found at 40,000-60,000 feet.
The speed difference between the jet stream air and the surrounding air, which can be as much as 100 mph, creates friction that leads to turbulence.

The SA flight turbulence occurred at around 37 000 ft. That's 3000 ft away from the jet stream - but it may still be close enough to be in the friction zone, i guess.