How do I solve these questions regarding homomorphism?
Notice the abuse of notation used. The quote overload symbol $f$ with three meanings which I will now denote differently with $f,f',f''$.
First is map from alphabet to words $$f: \Sigma \rightarrow \Gamma^*$$.
The second is the map between words $$f':\Sigma^* \rightarrow \Gamma^*$$.
The third is the map between languages $$f'':\mathbb{P}(\Sigma^*) \rightarrow \mathbb{P}(\Gamma^*)$$.

The The main point of the quote is that given $f$ there is the particular way to construct the corresponding $f'$ and $f''$. So, what one might call homomorphisms between languages $f''$ is specified by just giving map from alphabet to words $f$. And $f$ is usually easier to describe directly than $f''$. 

Another important observation is that not all function of type $$\mathbb{P}(\Sigma^*) \rightarrow \mathbb{P}(\Gamma^*)$$ is a homomorphism of languages. (Because it doesn't "preserve structure" w.r.t. concatenation operation languages are equipped with.) However, the said construction is special in that given any $f$ with the correct type, $f''$ will be a valid homomorphism of languages.