"Quasiparticles" are ubiquitous in condensed matter physics, e.g. magnons and phonons, and more generally all particles in quantum field theory are considered the elementary harmonic excitations ("vibrations") of the quantum fields. The classical analogues of these concepts are plane-wave excitations which are manifestly non-local (again, thinking of magnons and phonons as examples). Although localized instanton solutions do exist in classical gauge theories for example, they are not considered to be the classical analogues of particles in QFT's. So in what way can we see that the low-energy modes of a quantum field should be particle-like? Of course we always have particle-wave duality (which to me still feels like a phenomenological property rather than something we "see" at the level of field theory, say), but classically there are only waves, and there are no particles. Why can the harmonic (i.e. low-energy, quadratic) quantum fluctuations of fields be thought of as particles localized in space? What is the "extra ingredient" that quantum theory adds to the classical picture that "localizes" excitations?
There is a collection of concepts here that do not necessarily have anything to do with each other. Quasi-particles in condense matter physics and particles in quantum field theory are different things. The different quasi-particles are caused by the diverse dynamics that one finds in condense matter scenarios. They usually have a finite size and are seldom point-like.
On the other hand, the fundamental particles in quantum field theory are believed to be point-like. However, one needs to distinguish between the concept of a point-like particle and the elementary excitations of a quantum field. These two thing are not necessarily the same thing. Often when people refer to a single-particle state, they are referring to its wave function, which would in general not be point-like or localized in any way. The point-like particle nature is not given by any localization of such a single excitation. In fact, we cannot say anything about the point-like existence of particles unless we perform measurements. As a result, we can argue that it may be the measurement process itself that causes the point-like nature. It is the interaction of the field with measurement apparatus that exchanges a quantum of the field al a Planck's relation at a given localized point that gives this point-like observation of the particle. On the other hand, the natural evolution of quantum field without measurements is better described by the evolution of a wave function as given by the equations of motion and has nothing to do with point-like particles.
In homogeneous system the eigenstates of the Hamiltonian are typically plane waves and therefore not localized nor point-like. However, we can construct localized quasiparticle excitations out of them when the Hamiltonian is simple enough. One can simply form new eigenstates out of these plane waves by taking a linear superposition. When this is the case, one can form "wave packets" (typically a Gaussian superposition) which are localized and have a finite size. When is the Hamiltonian simple enough? At least in the cases where it contains no interactions: this is when the superposition principle holds and the wave packet can be easily constructed. Another way to phrase this is to say that the equations of motion are linear, as opposed to non-linear. In the non-linear case (i.e. when Hamiltonian has interactions) the different components of the wave packet will influence each other and scatter off from each other, creating a mess.