This phenomenon follows directly from the principle of time dilation of special relativity:

Proper time = time before time dilation

Observed coordinate time = time after time dilation

That means in this case: When each twin observes his own clock, the observed coordinate time is the proper time (time dilation factor 1, that means absence of any time dilation). When he observes some other clock moving at a relative velocity with respect to himself, the time dilation is not one, it is bigger than one, that means that there is some time dilation.