Why do we observe particles, not quantum fields? My understanding is that, in the context of quantum field theory, particles arise as a computational tool. We perform an expansion in the path integral in some parameter. The terms in these expansions correspond to Feynman diagrams which can be interpreted as interactions between particles.
My question: if particles are just a computational tool, why does it seem like particles really exist, and not quantum fields. For example, it seems like chemical reactions really do occur through the exchange of discrete particles called electrons. Are states with definite particle number somehow more probable, more easily observed, or some kind of classical limit?
 A: It isn't true that particles are just computational tools. The SLAC did not, for example, fire computational tools down a 10,000-foot beamline and measure their scattering angles and energies as they interacted with other computational tools ;-).
As commented by others, what we call a particle (in a beamline) is an excitation of a quantum field and in the world we inhabit, it is those particles that are manifest; we learn about the underlying field by the study of its excitations.
A: We don't observe particles, at least not in the sense of the physical definition of a particle (as the physical approximation of the motion of an extended classical body by the motion of its center of mass) or corpuscle (tiny pieces of matter).
What we are observing are quanta. Quanta are combinations of energy, momentum, angular momentum and charges (electric charges, lepton number etc.). These quanta are being irreversibly exchanged between quantum fields and external systems, like the detectors at CERN, for instance.
Quanta are not computational tools. They are the actual physical quantities that we are measuring in detectors and they differ in nothing from the classical energy, momentum, angular momentum and charge concepts.
What trips up many students and laypeople is the fact that quanta are properties and not objects. The "particle" nomenclature is one of the more unfortunate ones in physics. It suggests that quantum fields are made up of atomistic elements. That is not so. A general quantum field state does not have a fixed number of quanta that exist independently of emission and absorption processes. The  quanta we emit into a quantum field are in general also not the same as those that we absorb from the quantum field. Both of those simplifications exist only in the most trivial scenarios. In reality what we "emit" and "absorb" depends on the physical properties of the emitter and absorber and the physical interactions in the "free field", just like in non-relativistic quantum mechanics where we have to specify the initial state (and by that the properties of the system that does the "preparation" of the quantum state), the free dynamics and the measurement system (i.e. the specific type of absorber). Only if we have all three components defined do we have a description of a realistic physical system.
A: "My understanding is that particles arise as a computational tool."
This is a "cart before the horse" statement. Experiments determined that particles existed,( see here a bubble chamber photo of particle tracks) then "computational tools" were found by mathematically wise physisists that could model  the interactions of the observed particles.
Quantum fields are analogous to a coordinate system on which with differential operators on the named fields ( electron, neutrino...) one can mathematically model  the real world existence and interactions  of particles and check the validity of the model by comparing to data.
A: This is more of a side comment, but one way of thinking about quantum field theory is to regard it as a computational tool to compute particle interactions a la Weinberg : see https://arxiv.org/abs/hep-th/9702027 for this viewpoint on QFT. In a nutshell, we construct quantum fields to describe particle interactions obeying certain properties (e.g. Lorentz invariance, unitarity, etc) we expect to be held in realistic systems. Note that other answers in this thread seem to be using quanta as the terminology for particles.
Of course, there are cases in QFT where the concept of a particle doesn’t make sense; the scale-invariant theories or conformal field theories. In such a setting we actually measure correlation functions, as is typical of critical phenomena considered in condensed matter systems.
A: The theorists have mathematical rationalizations, but the reality (as Bohr pointed out) is that if your experiment senses fields you observe fields, while if it senses particles, you observe particles. Every radio detects the electromagnetic field, not photons. Waves in any field may be observed to diffract so long as their wavelength is accessible to a diffraction structure. On the other hand, if your experiment tracks particles, you'll observe particles.
