Composition of two continuous functions is continuous
Because $\delta'$ was chosen so that whenever an element $y$ of the domain of $g$ is $\delta'$-close to $b$, i.e.
$$|y-b|<\delta' $$ then the corresponding images under $g$ are automatically $\epsilon$-close, i.e.
$$|g(y)-g(b)|<\epsilon $$ because of the continuity of $g$.
Here, the two domain elements of $g$ are $\underbrace{f(x)}_{y}$ and $\underbrace{f(a)}_{b}$, so the guaranteed result is that
$$ |g(\;\underbrace{f(x)}_{y}\;)-g(\;\underbrace{f(a)}_{b}\;)|<\epsilon $$