Well, although I agree that the only concise picture of nature is given by mathematical models, it is always good advice, to keep an eye on what's measured that's described by the mathematical model. That's most important for QFT.
QFT has a very broad range of applicability. It's the most comprehensive mathematical model we have about nature. The only part that's not satisfactorily covered is gravity, but to discover the quantum theory of gravity is hindered by the fact that we have no observables concerning gravity that would need a quantum description. The classical limit of a (hopefully existing) quantum theory of gravitation is General Relativity, which is a classical field theory.
Relativistic QFT has been discovered by looking at high-energy particle-scattering experiments, and the physical quantities that are observable are corresponding cross sections. In the QFT formalism these are encoded in S-matrix elements which we are able to calculate in renormalized perturbation theory, which is well organized in terms of Feynman diagrams (sometimes with the necessity to resum infinitely many diagrams to get finite results; or using renormalization-group equations to improve the applicability of perturbation theory etc.). Thus, the physical meaning of the formalism is in the S-matrix elements, and to keep this in mind is very important, if you want to understand why relativistic QFT is formulated as it is. In the relativistic realm you need QFT, because particles are usually created and destroyed in scattering processes, which makes the formalism with creation and annihilation operators a very natural way to describe what's going on. The interpretability in terms of "particles" is very limited, since this notion makes only sense in terms of S-matrix elements, i.e., for asymptotic free states.
Another application of QFT in both relativistic and non-relativistic physics is many-body theory, i.e., the theory of very many particles building "matter" in the literal sense. There you can use QFT to derive macroscopic (semi-)classical equations to describe effectively the relevant (macroscopic) degrees of freedom encoded in material parameters like particle densities, energy-momentum densities (energy-momentum-stress tensor), various transport coefficients, etc. All these quantities are given in terms of correlation functions of field operators, often evaluated in thermal equilibrium (usually in the grand-canonical ensemble).