From spins to fields

In statistical field theory, one usually considers the so-called Landau Hamiltonian: $$\beta H = \int d^{d}x\bigg{[}\frac{t}{2}m^{2}(x) + \alpha m^{4}(x)+\frac{\beta}{2}(\nabla m)^{2}+\cdots+ \vec{h}\cdot \vec{m}(x)\bigg{]}$$ This Hamiltonian seems to be general enough to study Landau's theory on phase transitions. The process of constructing the above Hamiltonian is to coarse-grain the spin system and consider the order parameter (in my case, the magnetization) as a field. I'd like to better understand the motivations of such procedure: why to turn spin systems into fields? Does it improve our results? Does our models become more realistic this way? What kind of objects are we aiming to study with this field theory? And so on.