How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?

The given definition of $\operatorname{P\Gamma L}(V)$ is not the usual one: Typically $\operatorname{P\Gamma L}(V)$ is not defined as $\operatorname{\Gamma L}(V)/Z(\operatorname{\Gamma L}(V))$, but as the quotient group $\operatorname{\Gamma L}(V)/Z(\operatorname{GL}(V))$. Now it is clear that $\operatorname{PGL}(V)$ is a subgroup of $\operatorname{P\Gamma L}(V)$.

The slightly subtle point is that the center $Z(\operatorname{\Gamma L}(V))$ is not the same as the center $Z(\operatorname{GL}(V))$. The latter one is given by all diagonal maps $v\mapsto \lambda v$ with $\lambda\in F^\times$. However, for the center of $\operatorname{\Gamma L}(V)$ we get the additional condition that $\sigma(\lambda) = \lambda$ for all field automorphisms $\sigma\in\operatorname{Aut}(F)$. So $Z(\operatorname{\Gamma L}(V))$ only consists of the scalar matrices whose diagonal entry lies in the unit group of the prime field of $F$.

Given the way how $\operatorname{PGL}(V)$ is defined ("take the group and mod out its center"), the definition of $\operatorname{P\Gamma L}(V)$ might come as a surprise. The motivation for this slightly unexpected definition the is the following:

The natural action of $\operatorname{\Gamma L}(V)$ on $V$ induces an action on the subspace lattice of $V$. However, this action is not faithful. The kernel of the action is given by $Z(\operatorname{GL}(V))$. Modding out the kernel, we arrive at the group $\operatorname{P\Gamma L}(V)$ which acts faithfully on the subspace lattice of $V$. Now by the fundamental theorem of projective geometry, the so-defined group $\operatorname{P\Gamma L}(V)$ is precisely the automorphism group of the subspace lattice of $V$.

Conclusion

The right way to look at the transition from the groups $\operatorname{GL}(V)$ and $\operatorname{\Gamma L}(V)$ to the projective variants $\operatorname{PGL}(V)$ and $\operatorname{P\Gamma L}(V)$ is: Look at the natural action on the projective geometry (aka the subspace lattice of $V$) an mod out the kernel. This is also reflected by the letter "P" for projective in the nomenclature of those groups.