Suppose $S$ is an immersed submanifold of $M$. Let $\iota: S\hookrightarrow M$ be the inclusion map. Since it is an immersion, at each $p\in S$, $\iota_{*}:T_pS\rightarrow T_pM$ is injective. Hence we identify $T_pS$ with its image under $\iota_{*}$ . Under this identification, $\iota_{*}$ is the inclusion map
Recall that elements in $T_pM$ are defined by derivation on the space of germs of smooth functions at $p$.
I am not quite sure I am completely convinced with the bolded statement. May someone elaborate?
Your question about the bold statement does not really have to do with manifolds or geometry. It is just a general mathematical thing.
When you have an injective function $f \colon A \to B$ between sets (they don't have to be manifolds or tangent spaces, they can just be any sets), then since $f$ is "one-to-one", there is a bijection between $A$ and the image of $f$. It is common in math when there is a bijection to say $A$ when you really mean the image $f(A)$.
This is part of what bijections are all about. They allow you to identify two sets.