I have some (not quite finished) notes that hopefully explain some of this: http://web.mit.edu/~shopkins/docs/rsk.pdf.
By the way, I believe you got the map slightly wrong. It should be: $$ \left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \frac{bc}{b+c} & ab \\ ac & ad(b+c) \end{matrix} \right). $$ See Example 7 from the above linked notes to see why the tropicalized version of this map $$ \left( \begin{matrix} a & b \\ c & d \end{matrix} \right) \mapsto \left( \begin{matrix} \mathrm{min}(b,c) & a+b \\ a+c & a+d+\mathrm{max}(b,c) \end{matrix} \right) $$ is classical RSK in disguise. I should say I learned all of this material from Alex Postnikov.