Growth Rate of Solow Model
I do not directly see what intuition you are supposed to get from this growth rate equation, and the resulting math exercise but here is a sketch of the proof.

First determine the steady state capital by setting $\Delta K_{t+1}=0$ and solving for $K^*$. 

Substitute this value for the $(K^*)^{(2/3)}$ in the fraction. Replace the $ \frac{Y^*}{K^*}$ in the same expression with the value that you get capital-output steady state ratio, and work the $d$ into the parentheses.

The expression you have now can be shown to be the same as the growth rate of $y_t$ that can be found from the production function.