A good way to understand a particle in quantum field theory is through an analogy with solid state (condensed matter) physics.
Imagine a solid, there is a lattice of atoms, with a regular spacing. But the atoms can move, if one is displaced towards its neighbor, that neighbor moves a bit away thus influencing its neighbor and so on. So there is an underlying structure, the stationary lattice. And there are disturbances to that underlying structure, these various modes of vibration. You could imagine a small vibration, or a larger vibration. These vibrational modes are called phonons, and that's not a typo, they are like particles of sound. They aren't fundamental particles, they are modes of vibration of the lattice. But they obey certain rules, particularly if we look at the quantum nature of the lattice interactions.
Now instead of a lattice of atoms, imagine a single electron-positron field, filling all of space. It might have a vacuum state, and maybe there are disturbances or modes of vibration of that vacuum state, some of these modes are called electrons, some are called positrons. Some have momentum in one direction, some in others. Some have a spin of up, some of spin down. Each is a mode in the single unified electron-positron field that fills spacetime. Just as the lattice vibrations had quantized modes and we called them phonons, so these modes of the electron-positron field are quanta as well and they are electrons (or positrons).
If you look at a bunch of masses connected by springs, the normal modes are not localized at all. You can actually compute the eigen-modes for a 1d system of a few identical masses with identical springs, and it's a good exercise if you've never done it before. Similarly, the modes of a quantum field are not localized in the slightest and the field itself is everywhere.
It's similar to a wave having a Fourier transform, or a periodic wave having a Fourier series. When it has a Fourier version you can describe it as "made up" of a bit of this mode, and a bit of that mode. If later you have a very different Fourier transform then you can talk about dome of those bits having been destroyed or created. None of the parts, none of the modes, none of them are localized.
In QFT a particle is sometimes viewed as an irregularity in the field
If you think of the lattice as regular, then those vibrational modes can be thought of as an irregularity, but they are not localized in the slightest. One mode of vibration could be to mentally group the atoms into pairs of partner neighbors that then keep their center of mass constant as they move symmetrically towards their partner then away from their partner in a regular periodic fashion. That mode is not localized in the slightest. All the modes are qualitatively like that, they can be thought of as deviations from the regular lattice but they are deviations with their own regularity and are spread out everywhere, just as the original lattice was spread out everywhere.
So for QFT, instead of a lattice you have a vacuum state of a quantum field (such as the electron-positron field). The vacuum state is not zero, it is not trivial, and it has regularities. It is spread out everywhere. The higher energy modes of excitation are like deviations that themselves are regular and spread out over all space, just like the vacuum state was regular and spread out over all space. And while you are used to thinking of there being many electrons, there is only one field for every electron in the universe (and all the positrons share that field too). It is just allowed to have many excitations at once, like how a vibrating string can have many harmonics at once.
QFT treats particles as excited states of an underlying physical field, so these are called field quanta
The exact same thing, there is an underlying field, it can be excited in many ways, the excitations are what we call particles. They are like modes of vibration or deviations of the vacuum state. And all of them are spread out over all space.