There is no retraction map from unit disk to its boundary.
I was reading this proof:
It is found here in this link Retraction map from unit disk to its boundary
But I do not understand why the map $i_{*}$ is an injection, could anyone explain this for me please?
Also if someone could write for me a more clear proof than this, it will be greatly appreciated.
Your hypothesis is that there is a section $i$ of $r$, that is, $r \circ i$ is the identity $1_{S^1}$ of $S^1$. Then it holds that $(r \circ i)_*=(1_{S^1})_*$. Moreover, the following two equalities hold:
$(1)$ $(r\circ i)_*=r_* \circ i_*$
$(2)$ $(1_{S^1})_*=1_{H_1(S^1)}$
Indeed, the functional relation sending a continuous map $f\colon S \to T$ to $f_* \colon H_1(S) \to H_1(T)$ is a functor and the request that $(1)$ and $(2)$ hold is precisely the definition of the notion of functor.
It follows that $r_* \circ i_*=1_{H_1(S^1)}$. Hence $i_*$ is injective.