First, note that the result states that the only beta redex where the right-hand side is equal (modulo alpha-conversion) to the left-hand side is $(\lambda x. x x) (\lambda x. x x)$. There are other terms that reduce to themselves, having this redex in a context.
I can see how most of Lercher's proof work, though there are points where I can't get past without modifying the proof slightly. Suppose that $(\lambda x. A) B = [B/x] A$ (I use $=$ for alpha equivalence), and as per the variable convention suppose that $x$ does not occur free in $B$.
Count the number of $\lambda$'s in the left-hand side and the right-hand side. The reduction removes one from the redex, plus those of $B$, and adds as many as there are in $B$ times the number of occurrences of $x$ in $A$. In other words, if $L(M)$ is the number of $\lambda$'s in $M$ and $\#_x(M)$ is the number of free occurrences of $x$ in $M$ then $1 + L(B) = \#_x(A) \times L(B)$. The only solution to that Diophantine equation is $\#_x(A) = 2$ (and $L(B)=1$ but we won't use that fact).
I don't understand Lercher's argument for the paragraph above. He counts the number of $\lambda$'s and atomic terms; let's write this $\#(M)$. The equation is $\#(B) + 1 = \#_x(A) \times (\#(B) - 1)$, which has two solutions: $\#_x(A)=2, \#(B)=3$ and $\#_x(A)=3, \#(B)=2$. I don't see an obvious way to eliminate the second possibility.
Let us now apply the same reasoning to the number of subterms equal to $B$ on both sides. The reduction removes one near the top, and adds as many as there are substituted occurrences of $x$ in $A$, i.e. 2. Hence one more occurrence of $B$ must disappear; since the ones in $A$ remain (because $B$ contains no free $x$), the extra occurrence of $B$ on the left-hand side must be $\lambda x. A$.
I don't understand how Lercher deduces that $A$ does not have $B$ as a subterm, but this is not in fact relevant for the proof.
From the initial hypothesis, $[(\lambda x. A)/x] A$ is an application. This cannot be the case if $A = x$, therefore $A$ itself is an application $M N$, with $\lambda x.M N = [(\lambda x. M N)/x] M = [(\lambda x. M N)/x] N$. Since $M$ can't have itself as a subterm, $M$ cannot have the form $\lambda x.P$, so $M = x$. Similarly, $N = x$.
I prefer a proof with no counting arguments. Suppose that $(\lambda x. A) B = [B/x] A$.
If $A = x$ then we have $(\lambda x. A) B = B$, which is not possible since $B$ cannot be a subterm of itself. Thus, since the right-hand side of the hypothesis is equal to an application, $A$ must be an application $A_1 A_2$, and $\lambda x. A = [B/x] A_1$ and $B = [B/x] A_2$.
From the former equality, either $A_1 = x$ or $A_1 = \lambda x. [B/x] A$. In the second case, $A_1 = \lambda x. (\lambda x. A_1 A_2) B$, which is not possible since $A_1 cannot be a subterm of itself.
From the latter equality, either $A_2 = x$ or $A_2$ has no free $x$ (otherwise $B$ would be a subterm of itself). In the latter case, $A_2 = B$.
We have shown that $A = x x$. The right-hand side of the initial hypothesis is thus $B B$, and $B = \lambda x. A$ = $\lambda x. x x$.