Context-free Languages closed under Reversal
Your sources are right, and I am afraid there is only little to add, except formalism. I denote the reverse (mirror) of string $w$ by $w^R$.

If $G$ is a grammar, let $H$ be its reversed, so for production $A\to w$ in $G$ we have $A\to w^R$ in $H$. 

Then by induction we show that $A\Rightarrow_G^*w$ iff $A\Rightarrow_H^*w^R$. 


(basis) In zero steps we have  $A\Rightarrow_G^0 A$ iff $A\Rightarrow_H^0 A$. 
(induction) Assuming
$A\Rightarrow_G^*w_1Bw_2$ iff $A\Rightarrow_H^*w_2^RBw_1^R$ we can apply any production $B\to u$ in $G$ (and in $H$ in reverse) and obtain $A\Rightarrow_G^*w_1uw_2$ and $A\Rightarrow_H^*w_2^Ru^Rw_1^R$ respectively, where indeed $w_2^Ru^Rw_1^R$ is the reverse of $w_1uw_2$.


This is a very condensed proof, but contains all necessary ingredients. Again, a derivation of the reverse grammar is the reverse of the original one. This is especially clear when looking at the two derivation trees.