It claimed that 'cats domesticated themselves'. But what does it mean? Firstly the statement should be amended to read, 'wildcats domesticated themselves'. That makes more sense and is more precise.
Specifically the African-Asian wildcat domesticated itself. But I am unsure if this is correct. The process of domestication takes two parties: the animal and the human. They play at least an equal role and perhaps in this instance the human played the major role.
What it means is that wild cats liked to be around farmers' grain stores about 10,000 years ago because there were hordes mice and other rodents there providing a supply of food. It was very attractive to a wild cat and it still is.
You'll still see wild cats near human settlements in Africa which is why they become hybrids having matted with domestic or semi-domestic cats.
Anyway because the cats willingly came to the farmers and hung around allowing the humans to get to know them it is said that the cats domesticated themselves.
The better description would be that they allowed themselves to be domesticated as a consequence of being in contact with people. And of course the people desired that wild cat domestication. A two-way process.
The leader in the process must be the human. It is still the same now in the human-domestic cat relationship. The human leads and largely dictates the terms of the friendship.
Although there is a large amount of training that goes on by the domestic cat of the human. It is a gradual, informal version of training but equally effective but it once again relies on the agreement and cooperation of the owner.