Möbius transformations on the complex plane are functions of the form $$z\mapsto\frac{az+b}{cz+d}\qquad\hbox{for }a,b,c,d\in\mathbb{C}, ad-bc\neq 0$$
Any Möbius transformation can be created by composing three primitive types of transformation: scaling by a constant, translating by a constant, and inversion (mapping $z\mapsto 1/z$).
Less obviously, you can manage without the scaling primitive, because you can make a scaling out of translations and inversions! You need three of each: invert, translate by $a$, invert, translate by $-1/a$, invert, and finally translate by $a$ again. If you write down those six maps, compose them, and simplify frantically, this works out to a scaling by $-a^2$: $$\bigl[(z\mapsto z+a)\circ(z\mapsto{\textstyle\frac1z})\circ(z\mapsto z-{\textstyle\frac1a})\circ(z\mapsto{\textstyle\frac1z})\circ(z\mapsto z+a)\circ(z\mapsto{\textstyle\frac1z})\bigr](z)\enspace=\enspace\frac1{\frac1{\frac1z+a}-\frac1a}+a\enspace=\enspace-a^2z$$ And since we're working in $\mathbb{C}$, everything has a square root, so there's always an appropriate $a$ for any value you want to scale by.
This web toy lets you play with this concept, by showing a starting square and its six images under those successive translations. The red square is the starting square, and the arrows indicating the successive transformations connect the squares in order, ending with the final square scaled by $-a^2$ (in purple). You can drag the starting square around, or resize and rotate it by dragging one of its vertices.
You can also drag any of the three circles connected to the origin by arrows. These represent the various constants: