The notation of $splits(label)$ under Random Forest
You correctly mentioned the definition of impurity which is
$$I_G(P) = 1 - \sum _{i=1}^k p_i^2$$
This can be written as
$$I_G(P) = \sum _{i=1}^k p_i*(1 - p_i)$$
At any split, for any branch $T_i$, you calculate the probability using the classical definition i.e.,
$$p_i = \frac{|T_i|}{|T|}$$
Using this, you can derive the impurity definition $I(T_i)$ given in the paper.
Once you are able to calculate the impurity of a branch you can calculate the impurity of a node using the weighted sum of each branch, where weights are the probability of that branch. Hence,
$$I_{node} = \sum_{\forall T_i  \in splits(a)} \frac{|T_i|}{|T|} I(T_i)$$
Using the above impurity, you can calculate the change in impurity or the information gain as given in the paper.
I hope this solves your doubt.