How to apply "Substitution Method" in Difference Equations?
Substitution is not a method to obtain the particular solution $y_p(n)$, but the author simply claims that $y_p(n)=e^{-\beta n}u(n)$ is a particular solution of the recursion (1.70), and encourages you to check his claim by substituting $y_p(n)$ for $y(n)$ in Eq. (1.70) and see if it's correct or not:

$$e^{-\beta n}u(n)=e^{-\beta}e^{-\beta (n-1)}u(n-1)+\delta(n)=e^{-\beta n}u(n-1)+\delta(n)\tag{1}$$

Now check if (1) is satisfied. Distinguish 3 cases: $n&lt;0$, $n=0$, and $n&gt;0$. As a result you should be able to see that (1) (and, therefore, Eq. (1.70)) is satisfied for $y(n)=y_p(n)$, showing that the suggested sequence $y_p(n)$ indeed is a particular solution of (1.70).