From Rational Points on Elliptic Curves by Silverman and Tate
To obtain Weistrass Normal form, it uses some technique of changing coordinates, but I can't understand its description, I underlined them by red lines.
What does those lines mean? For example, What is the relation between "coordinate changing" and "taking $Z=0$ to be the tangent(I don't know what this mean)"? What coordinate change should I make? Can someone explain it to me and give some examples to illustrate its step?