Whiskering a natural isomorphism yields a natural isomorphism

$\alpha : F \Rightarrow G$ is a natural transformation and $X$ is some object of $\mathscr C$: remember that a natural transformation is a family of morphisms, indexed over the objects of the domain, so really we're parameterising over all objects $X \in \mathscr C$. The whiskering $J \alpha : \mathscr C \Rightarrow \mathscr E$ is also a natural transformation, which is formed by applying $J : \mathscr D \to \mathscr E$ to each $\alpha_X : FX \to GX$ to get a family of morphisms $J (\alpha_X) : JFX \to JGX$. We usually simply write this as $J \alpha_X$.

Now let us assume $\alpha : F \Rightarrow G$ is a natural isomorphism, which means that each morphism $\alpha_X : FX \to GX$ is an isomorphism. The whiskering $(J \alpha)_X : JFX \to JGX$ is formed by applying $J$ to each $\alpha_X$. As you point out, applying functors preserves isomorphisms (more generally, they preserve commutative diagrams), so if $\alpha_X$ is an isomorphism, then so is $J(\alpha_X)$. This is the statement of Theorem 107: whiskering a natural isomorphism by any functor gives you another natural isomorphism.

Here $X$ is an object of $\mathcal C$.
Remember that a natural transformation [isomorphism] $\alpha:F\to G$ is a collection of arrows [isomorphisms] $\alpha_X:F(X)\to G(X)$ for $X\in Ob\,\mathcal C$ (the 'components of $\alpha$'), satisfying a commutativity condition for each arrow in $\mathcal C$.

So, the components of $J\alpha$ are $J(\alpha_X)$, which are isomorphisms if all $\alpha_X$ are.

Similarly, the components of $\beta F$ are $\beta_{F(X)}$.

Finally, you would want to convince yourself that these collections indeed satisfy the commutativity conditions.