How did cats become domestic animals
The domestication process developed gradually. It might have started with the most peaceful cats and those who liked men the most. In those times, whelps must have already awaken sympathies and loving care from those who loved these kind of animals. Closeness between cat and man and their constant care for them, brought the idea to select the most peaceful specimens.
Romans brought them to Europe: When Romans spread the Egyptian domestic cat throughout Europe, breeding must have happened with local mountain wild cats (fells silvestris). When, after the crossing, wild features were evidenced in young cats, these were not chosen to be domesticated. So, in spite possible biological combinations regarding wild characteristics of both species, along with the years, our domestic cats must have inherited just a minimum part of them.
On the opposite case, young mountain cats would have not been able to survive in the wild carrying domestic cat's characteristics. This is the explanation of why these two species have survived until now.