Why we call the identity map the inclusion? In the following picture :

The hint beside question 2(a) said that we will use $g$ as the inclusion map, while in the following picture the solution is given by considering $g$
  the identity map.

What is the relation between the identity map and inclusion map?
 A: For example, the map $f:\mathbb{Z} \to \mathbb{Q}\;$defined by
$$f(n) = n,\;\text{for all}\;n \in \mathbb{Z}$$
is an inclusion map, but is not an identity map. 

For one thing, it's not surjective. 

More importantly, the domain and codomain are not the same.

An identity map is a function from a set $A\;$to itself which fixes every element of $A$.

An inclusion map is a function from a set $A\;$to a set $B,\;$where $A \subseteq B,\;$which fixes every element of $A$.
A: The identity map and the inclusion map are both injective. If $A\subseteq B$, we can define the inclusion map $i:A\longrightarrow B$ from a set $A$ to a set $B$ as follows:
$$i(x)=x,\ \forall x\in A$$
this is a generalization of the identity map which correspond to the case $A=B$. Both are injective maps.
In your text, the inclusion is present because $D=\mathrm{ker}(f)\subseteq A$. So, you can consider an monomorphism (that is, injective homomorphism) given by the inclusion map:
$$g:D\longrightarrow A$$
and this defined by:
$$g(x)=x,\ \forall\, x\in D.$$
I call $g$ the inclusion map of $D$ in $A$ like your text.
