For a map $f: X \to Y$, we let $[f]$ denote the equivalence class containing $f$, called the homotopy class of $f$. Therefore, since homotopies are compatible with composition, it follows that if $\alpha = [f] \in [Y,Z]$ and $\beta = [g]\in [X,Y]$ then $\alpha \beta = [fg] \in [X,Z]$.
I am confused by this claim, because it seems to be implying that the homotopy classes $\alpha$ and $\beta$ are homotopic, but I don't see how that is possible.
For me what is written is just a definition : you have $\alpha\in [Y,Z]$ and $\beta\in [X,Y]$, and you want to define $\alpha\beta$.
The first thing that comes to mind is that is $\alpha = [f]$ and $\beta = [g]$, then we define $\alpha\beta = [fg]$. But a priori this is ill-defined, you have to check that it doesn't depend on the choice of $f$ and $g$, ie that if $[f] = [f']$ and $[g] = [g']$, then $[fg] = [f'g']$. And there is a (basic) theorem that says this is true, which is referred to as "since homotopies are compatible with composition" in your text.