Transformation from one feature space to another
This is just the explicit form of the feature map for a homogenous, degree 2 polynomial kernel.  Here your input space is $\mathbb{R}^2$ and the feature space can be identified with $\mathbb{R}^3$.  The point the author is making is that instead of computing the feature map explicitly, and then taking the Euclidean inner product $\langle \Phi(p),\Phi(q)\rangle$, you can just compute $\langle p,q\rangle^2$.