Understanding derivation of scattering from Maxwells eqns...

I am trying to follow a Maxwell's equations derivation for light scattering but don't understand 'why' the authors do the steps they do at this start bit. Help would be greatly appreciated...

It starts with the incident electric field equation.
[tex]\textbf{E}_{0}(\textbf{r},t) = \textbf{E}_0 \exp \left[i \left(\textbf{k}_0 \cdot r-\omega t\right)\right] \qquad [1][/tex]
Then they show
[tex]\nabla \times \textbf{E}(\textbf{r},t) = -\frac{\partial}{\partial t}\textbf{B}(\textbf{r},t) \qquad [2] \\
\nabla \times \textbf{H}(\textbf{r},t) = \frac{\partial}{\partial t}\textbf{D}(\textbf{r},t) \qquad [3][/tex]
The scatterer is described by
[tex]\textbf{D}(\textbf{r},t) = \varepsilon(\textbf{r}) \cdot \textbf{E}(\textbf{r},t) \qquad [4] \\
\textbf{B}(\textbf{r},t) = \mu_{0} \textbf{H}(\textbf{r},t) \qquad [5][/tex]
Now this is where I get lost

"Taking the curl of eq.[2] using that
[tex]\nabla \times ( \nabla \times \textbf{E}) = \nabla ( \nabla \cdot \textbf{E}) - \nabla^{2} \textbf{E}[/tex] and substitutions of Eqns [3,4,5] yields a single equation for the total electric field strength,
[tex]\nabla(\nabla \cdot \textbf{E}(\textbf{r},t)) - \nabla^{2}\textbf{E}(\textbf{r},t) = -\mu_{0} \varepsilon(\textbf{r}) \cdot \frac{\partial^{2}}{\partial t^{2}} \textbf{E}(\textbf{r},t)[/tex]
So why do they just decide to take the curl of equation 2 using that identity? And then subbing all those equations in? What does it mean physically? I really need everything spelled out in very basic ways to help me understand.

Thanks