Speaking in a video from his hospital bed in Switzerland, the former Top Gear host detailed the knee injury he suffered in the incident which left the vehicle on its roof and scorched.
He said: "Hello, yes it's true I binned it again," before adding he will need surgery to fix his broken leg.
"I would like to thank all of the medical professionals who got me by air ambulance from the crash to the hospital and who have dealt with me ever since."
In the Drive Tribe video message, titled I'm Not Dead, Hammond adds: "And most importantly, sorry to my wife Mindy and to my daughters Issy and Willow."
The crash took place in Switzerland where the TV presenter, 47, was driving an electric super car, and his colleague Jeremy Clarkson described it as "the biggest crash I've ever seen and the most frightening".
A spokesman for The Grand Tour said: "Richard Hammond was involved in a serious crash after completing the Hemberg Hill Climb in Switzerland in a Rimac Concept One, an electric super car built in Croatia, during filming for The Grand Tour Season 2 on Amazon Prime, but very fortunately suffered no serious injury.
"Richard was conscious and talking, and climbed out of the car himself before the vehicle burst into flames.
"He was flown by Air Ambulance to hospital in St Gallen to be checked over, revealing a fracture to his knee.
"Nobody else was in the car or involved in the accident, and we'd like to thank the paramedics on site for their swift response.
"The cause of the crash is unknown and is being investigated."
In a post on Twitter on Sunday morning, F1 presenter Sasha Martinengo told his Twitter followers the pair had spoken and that Hammond required surgery.
He posted: "Hi all. Got a msg from @RichardHammond. He is fine. He has a broken leg and needs knee surgery. He might fly home for the procedure."
Clarkson tweeted: "It was the biggest crash I've ever seen and the most frightening but incredibly, and thankfully, Richard seems to be mostly OK."
A statement from the organisers of the Bergrennen, which translates as mountain race, said: "On Saturday, an accident took place at the Bergrennen Hemberg in a demonstration run between the training courses.
"The vehicle left the track and landed in the meadow."
They added that rescuers were "very fast" to reach the scene.
"The fire brigade extinguished the burning car quickly," they said.
It comes 11 years after Hammond suffered life-threatening head injuries following a high-speed crash as he filmed for BBC's Top Gear in 2006.
The presenter was in a coma for two weeks following the 288mph accident, but made a full recovery.
In March this year, Hammond assured fans he was fine after falling off a motorbike "many times" while filming for The Grand Tour.
He was in a remote part of Mozambique in east Africa shooting the programme when the accident occurred, according to The Sun.
Grand Tour producer Andy Wilman said the presenter was seconds from being "incinerated".
"Jeremy and James both raced down to the scene of the crash as soon as it happened," the Sunday Mirror said.
"When they saw the wreckage on fire they thought Richard was dead. It was really bad. They were staggered he had got out of it alive, because there was just nothing left."