How exactly is the Poincare group related to the free relativistic theories in quantum field theory? I know Poincare group is the Lorentz group along with translations but don't see any connected why the field will influence the Poincare group.
This is a general phenomenon. Whenever a theory/system $S$ has a symmetry, it in particular means that some group $G$ acts $G\times S \to S$ on the system $S$. The system $S$ then constitutes a (possibly non-linearly realized) representation of the group $G$. See also my Phys.SE answer here.
Concretely the isometry group of Minkowski space is the Poincare group. Concerning representations of the Poincare group in relativistic field theories, see e.g. this & this Phys.SE posts and links therein.