How are we able to use quantum field theory to study systems?
The same question could also be asked about quantum mechanics, or even about classical mechanics.
For example, in the Newtonian mechanics of gravitation, every mass in the universe instantaneously influences any test mass you are studying. In principle, you should take into account every mass in the universe when you study the dynamics of a marble rolling down an incline plane. But, in practice you can ignore everything except the earth and the marble.
As another example, in the quantum mechanics of scattering, introductory treatments often describe the scattering of "plane waves." Plane waves have a wave function that looks like this:
which extends over the entire universe and in fact has a probability of |A|^2 to be found anywhere in the universe (which is clearly not normalizable). In this case, the introductory treatments often elide the fact that the true scattering states need to be described by wave packets, which are localized wave functions with relatively precise positions and momenta. These localized wave still take on a value at every point in the universe, but they fall off to zero very quickly away from the locations of the particles being studied.
In quantum field theory, we demote the position from an operator to a parameter, and we promote the fields to operators. You might see the Dirac field operator describing the electron/positron field dynamics written like:
where the field takes on a value at every point in the universe. But, as with the above examples, in any scattering experiment, such as at LHC, the field description of the colliding particles involves localization.
To bolster this argument with an authoritative reference, we can look at Schwartz's "Quantum Field Theory and The Standard Model" at page 97, where he states:
"Constructing local theories out of fields made from creation and annihilation operators guarantees cluster decomposition, as we have seen."
This means, for example, when you compute the scattering S-matrix you only need to consider "connected" diagrams (connecting your in and out scattering states), and you don't need to consider interference from diagrams related to scattering taking place on the other side of the country.
However, Schwartz does provide the caveat:
"It is also not clear how well cluster decomposition has been tested experimentally."