What do we actually prove using induction theorem?

Induction is, intuitively, an outline of an infinite proof.

You first prove $P(0)$, the base case.

Then you prove $P(1)$ follows from $P(0)$.

Then you prove that $P(2)$ follows from $P(1)$.

Et cetera.

In general, if you know that $P(k+1)$ follows from $P(k)$, and you know that $P(0)$ is true, then you know how to prove $P(n)$ for any natural number. For example, $P(4)$ follow because $P(0)\implies P(1)\implies P(2)\implies P(3)\implies P(4)$.

So you don't need to know how to show $P(k)$ to show that $P(k)$ implies $P(k+1)$. Once you've shown this implication, and $P(0)$, we know how to write a non-inductive finite proof of $P(10000)$ or $P(10^{99})$.

Your book is right. You have to show that $$P_k \implies P_{k+1}$$

Because if you prove that and since you said it is true for base number, the proposition is automatically proved true for every $n \in S$.

Let us see how!

To prove "$P_k \implies P_{k+1}$" means you are proving that a particular proposition has a quality of being true for $k+1$ because of being true for $k$.

Improvement (Edit): Just as you do in case of domino effect. Suppose it is reported that the cornered domino has been pushed over. And then you find out that dominoes are arranged such that whenever $k^{th}$ domino falls, $(k+1)^{th}$ falls. You would at once say that all dominoes must fall.

Remember that these two conditions of induction theorem simultaneously imply that a certain proposition is true.

i) It is true for base number

ii) It is true for an integer $k$ implies it is true for integer $k+1$ after it.

For the proposition in your picture, if you would take base number $2$ then after doing all that (done in your book), you would prove that proposition is true for integer $2$ and integers after it. But since we usually ask to prove that conjecture for every $n \in \mathbb N$, we take base number $1$ not $2$ because the smallest number of set $\mathbb N$ is $1$.

(Wanted to post as a comment, but it is too long :/)

When we prove something by induction we prove that our claim is correct for a base case (for example, n=1). Afterwards we assume (not proving, only assuming) that our claim stands for some arbitrary value k and than, based on the assumption we prove it holds for k+1.

Note that based on the proof of base case and the proof that if $P_k$ is true we show that $P_{k+1}$ is true. Example: you prove that some claim is correct for $n=1$, then you assume it is correct for some n=k. Now you prove for n=k+1.

We proved for n=1 and using the assumption that the claim stands for n=k we proved for n=k+1. In this case, we know that the claim stands for n=1, thus we can now say it stands for n=2. Afterwards we can say that it stands for n=3 and so on, hence it stands for any n.

That is how we use induction - you don't prove that $P_k$ is true, you assume it it true - that is a big difference.

If my explanation is not coherent enough, you might want to google "Induction" and read a little more.

Good luck.