fractional linear transformation (sometimes called a Möbius transformation). These have the property that they take (lines and circles) to (lines or circles).
The boundary of the domain is the real axis. This is a line, so the mapping f will take it to a line or a circle. A circle is determined by three points on it, so take three points on the real line, say 0, 1 and –1, and see what f does to them. You find that f(0) = –1, f(1) = –i and f(–1) = i. The circle through the three points –1, i and –i is the unit circle.
Therefore f takes the boundary of D to the unit circle. The circle splits the complex plane into two regions, namely the inside and the outside of the circle. So f must take D to one or other of those regions. To find out which, choose a point in D and see where f takes it. If you take the point i, you see that f(i) = 0. From that, you conclude that f takes D to the inside of the unit circle, namely the open unit disc.
That method is not nearly as slick as the trick used by the tutor, but it is a standard method for keeping track of what a fractional linear linear transformation does to a half-plane or a disc.