Talk the hind legs off a donkey - that's a strange idea.
The hind legs are the back legs and somebody who could 'talk them off,' is very talkative indeed. Perhaps they could talk so much that they could persuade anybody to do anything.
So, in simple terms this idiom is used to refer to a very talkative person.
Here is a very nice video with a story called The talkative man based on the wonderful Magudi Days written by R.K Narayan - one of my favourite authors.
It will also give you practice at listening to Indian English.