I really can't wrap my head around how to prove things using Gentzen system, $\mathscr G$. Could someone try to explain how these proofs works, my book "Mathematical logic for computer science" is really hard to grasp. For example I would like to prove $\vdash (A \to (B \to A)) $ in a similar fashion to the examples in the following image.
A Gentzen system is, to my knowledge, a deductive proof system using sequents. In order to write a proof, you will need to know the inference rules and/or axioms of the particular system you are using.
Now the system in your book is a little different from what I am familiar with, but it seems to be based on propositional logic with the additional classification of formulae into "$\alpha$" and "$\beta$" formulae. Based on fig. 3.1 on pg. 51, $\alpha$-formulae are those formulae $A$ with subformulae $A_1$ and $A_2$ such that $A=A_1\lor A_2$ and $\beta$-formulae are those formulae $B$ with subformulae $B_1$ and $B_2$ such that $B=B_1\land B_2$.
The inference rules provided are:
Let $\{\alpha_1,\alpha_2\}\subseteq U_1$ and $U_1'=U_1\setminus\{\alpha_1,\alpha_2\}$.
Rule: $U=U_1'\cup\{\alpha\}$
Let $\{\beta_1\}\subseteq U_1$, $\{\beta_2\}\subseteq U_2$, $U_1'=U_1\setminus\{\beta_1\}$, and $U_2'=U_2\setminus\{\beta_2\}$
Rule: $U=U_1\cup U_2\cup \{\beta\}$
Here $U_1$ and $U_2$ are used to refer to sets of formulae in the premises, and $U$ is used to refer to sets of formulae in the conclusion.
Now, I'm not sure why the author wrote it this way. A much easier way to put this would be:
Let $P$ be a set of formulae and $p_1,p_2\in P$. Let $P'=P\setminus\{p_1,p_2\}$. From $P$, we may conclude $P'\cup\{p_1\lor p_2\}$.
e.g.:$\dfrac{\vdash a,b,c}{\vdash a, b\lor c}$
Let $P_1$ and $P_2$ be sets of formulae and $p_1\in P_1$, $p_2\in P_2$. Let $P_i'=P_i\setminus\{p_i\}:i=1,2$. From $P_1$ and $P_2$, we may conclude $P_1'\cup P_2'\cup \{p_1\land p_2\}$
e.g.:$\dfrac{\vdash a,b\qquad \vdash c,d}{\vdash a,c,b\land d}$
So, in your example, the steps, laid out one at a time, are:
$\dfrac{\vdash\neg p,q,p\qquad\vdash\neg q,q,p}{\vdash\neg p\land\neg q,q,p}$ from inference rule 2
$\dfrac{\vdash \neg p\land\neg q,q,p}{\vdash \neg(p\lor q),q,p}$ from $\neg (B_1\lor B_2)=\neg B_1\land\neg B_2$ (shown in fig 3.1)
$\dfrac{\vdash \neg (p\lor q),q,p}{\vdash \neg(p\lor q),q\lor p}$ from inference rule 1
$\dfrac{\vdash \neg (p\lor q),q\lor p}{\vdash \neg(p\lor q)\lor (q\lor p)}$ from inference rule 1
$\dfrac{\vdash \neg(p\lor q)\lor(q\lor p)}{\vdash(p\lor q)\to(q\lor p)}$ from $A_1\to A_2=\neg A_1\lor A_2$ (shown in fig 3.1)
How you prove $\vdash A\to(B\to A)$ depends on where you start.
The shortest proof that I can think of would be: $$\begin{align}1.& \vdash \neg A,\neg B,A & \text{Axiom}\\2. & \vdash \neg A,\neg B\lor A\qquad & \text{inference rule 1}\\ 3. & \vdash \neg A,B\to A & \text{$B_1\to B_2=B_1\lor B_2$ (shown in fig. 3.1)}\\ 4. & \vdash \neg A\lor (B\to A)& \text{inference rule 1}\\5. &\vdash A\to(B\to A)&\text{$B_1\to B_2=B_1\lor B_2$ (shown in fig. 3.1)}\end{align}$$