How to prove the structured program theorem?
The Böhm-Jacopini theorem essentially says that a program based on 'goto' is functionality equivalent to one consisting of 'while' and 'if'.

Sketch of proof based on the following lecture notes: 

Say you have a program that consists of a sequence of statements $S_i$; prefix each statement with a label $S_i^{\prime} \gets L_i: S_i$ and update existing goto statements to point to the right locations. Declare a location variable, $l \gets 1$, and wrap the prefixed statements in a while loop that will continue until the last statement is reached, $\textbf{while} (l \neq M) \ \textbf{do} \ S^{\prime}$. 

Apply the following rewrite rules to $S^{\prime}$:


Goto rule: Replace $L_i \ \textbf{goto} \ L_j$ with $\textbf{if} \ (l = i) \ \textbf{then} \ l \gets j$ 
If-goto rule: Replace $L_i : \textbf{if} \ \text{(cond)} \ \textbf{then} \ \textbf{goto} \ L_j$ with $\textbf{if} \ (l = i \land \text{(cond)}) \ \textbf{then} \ l \gets j \ \textbf{else} \ l \gets l + 1$
Otherwise rule: Replace $L_i : S_i$ with $\textbf{if} \ l = i \ \textbf{then} \ S_i,  \quad l \gets l + 1$


The resulting program is then free of goto statements showing the correspondence between the two paradigms.

A more formal proof is provided in your second reference.

Update 

After studying the paper in greater detail, this is my interpretation:

Based on the definitions in CACM, let $\mathcal{B}^{\prime} = \mathcal{D}(\alpha, \lbrace \lambda, R \rbrace)$ be the class of Turing machines represented by the language $\mathcal{P}^{\prime}$ that covers flow diagrams. Let $\mathcal{B}^{\prime \prime} = \mathcal{E}(\alpha, \lbrace \lambda, R \rbrace)$ be the class of Turing machines represented by the language $\mathcal{P}^{\prime \prime}$ that covers the transformation covered above.

According to CACM, the author wants to prove that if a Turing machine exists in the family of Turing machines described by $M \in \mathcal{B}^{\prime}$, then there exists a corresponding Turing machine in the family of Turing machines described by $M^{*} \in \mathcal{B}^{\prime \prime}$. 

Sketch of the author's proof: He does this by drawing from the derived relationship $M \in \mathcal{B}^{\prime}$ and $M \in \mathcal{E}(\cdots)$ (definition (5)) and then sets up the relationship $M^{*} \in \mathcal{E}^{*}(\cdots)$. For each component of $\mathcal{E}(\cdots)$ he draws a one-to-one mapping with those of $\mathcal{E}^*(\cdots)$. Thus by establishing the bridge between $M$ and $M^{*}$ through $\mathcal{E}(\cdots)$ the author proves $M \in \mathcal{B}^{\prime} \implies M^{*} \in \mathcal{B}^{\prime\prime}$.