The group $G(\mathfrak{P})$ is the collection of all $\sigma\in\mathrm{Gal}(L/K)$ such that $\sigma(\mathfrak{P}) = \mathfrak{P}$. So $G(\mathfrak{P})\cap\mathrm{Gal}(L/F)$ is the collection of all $\sigma\in\mathrm{Gal}(L/K)$ that fix $F$ pointwise and map $\mathfrak{P}$ to $\mathfrak{P}$ (as a set); i.e., the decomposition group of $\mathfrak{P}$ over $F$.
Likewise, the inertia group $I(\mathfrak{P})$ is the subgroup of all $\sigma\in\mathrm{Gal}(L/K)$ such that $\sigma(a)\equiv a \pmod{\mathfrak{P}}$ for all $a\in\mathcal{O}_L$. Intersecting with $\mathrm{Gal}(L/F)$ gives you the inertia group over $F$.
So, for example, the size of $G(\mathfrak{P})\cap\mathrm{Gal}(L/F)$ gives you the inertial degree of $\mathfrak{P}$ over $\mathfrak{P}\cap\mathcal{O}_F$ (and hence the inertial degree of $\mathfrak{P}\cap\mathcal{O}_F$ over $\mathfrak{P}\cap \mathcal{O}_K$); and the size of $I(\mathfrak{P})\cap\mathrm{Gal}(L/F)$ gives you the ramification index of $\mathfrak{P}$ over $\mathfrak{P}\cap\mathcal{O}_F$. Basically, any information you can generally get out of the decomposition and inertia groups, plus what information you can get out of the multiplicativity of the inertial degrees and ramification indexes in towers.