Boosting fits trees one at a time, each correcting what the ensemble so far got wrong. Chen & Guestrin's contribution is to make "what it got wrong" precise: take a second-order Taylor expansion of the loss around the current prediction, and every quantity you need falls out in closed form.

With $g_i$ and $h_i$ the first and second derivatives of the loss at the current margin, the regularised objective for a tree with leaves $j$ is:

Minimising over $w_j$ gives the optimal leaf weight and the objective it achieves:

And the gain from splitting a node into $L$ and $R$ is the drop in that objective:

This is not an impurity measure. It is a direct estimate of how much the loss will fall, which is why the Hessian appears in every denominator.