The creature - the last known surviving member of his species - passed away at Galapagos National Park in June at an estimated age of 100 years old.
Speaking at the launch of his new Sky 3D series about the Galapagos Islands, Attenborough explained that the third episode will feature his meeting with Lonesome George.
"He has caught the world's imagination because you can't be rarer than just the last individual [of a species or subspecies]," he said. "For years it was hoped we would be able to find a female for this one particular individual from this species - nobody did.
"Lonesome George was there sitting in a sanctuary in the middle of the island for people to go and see, but we never found a mate for him. We don't know how old he was, but it was a privilege to see him and we're grateful for the Galapagos authorities for letting us go in because he was an old gentleman, you know - I know how he feels!"
Attenborough joked that the crew were only allowed 30 minutes with Lonesome George, who spent most of the time asleep.
"There was old Lonesome George underneath his bush asleep, so the temptation of course was to say [knocks], 'I say, hello! Anybody in there?'" he quipped. "But that was not allowed. So we just sat there alongside until poor old Lonesome eventually stuck his head out and looked me straight in the eye, and I delivered my words.
"Then old Lonesome pushed off, creaking a bit, and then I think it was a fortnight later he died in his sleep. So we have a unique picture of old Lonesome George, for whom I retain a great affection really. It was a great story."
David Attenborough's Galapagos 3D premieres at 7pm on Sky 3D and Sky1 tonight (January 1).
Watch David Attenborough speaking to Digital Spy about Galapagos 3D below: