What is the use of knowing elasticity of average product? Firstly they have given output elasticity of a factor, use of which is clear to me. It says to what degree my total output will vary due to a change in the quantity of a factor.
But what else do we get to know from elasticity of Average product?
If you know Elasticity of average product you can identify how your output per worker or machine will change as $L$ or $K$ increases.

Recall the definition of elasticity is:

$$\mathcal{E}=\frac{\Delta y/y}{\Delta x/x}$$

average product of labor and capital:

$$AP_L=\frac{Q}{L}$$
$$AP_K=\frac{Q}{K}$$

therefore the elasticity of average product of labor and capital is:

$$\Gamma_L=\frac{\Delta AP/AP}{\Delta L/L}$$
$$\Gamma_K=\frac{\Delta AP/AP}{\Delta K/K}$$

This is useful for evaluating how existing worker or capital productivity will be impacted as a result of an increase in capital or labor.