Well, forget these handwaving explanations with "virtual particles" and "vacuum fluctuations". It's of course hard to study the real thing, which is quantum field theory, for which you need quite some pretty abstract math, but one can also try to explain the issues without these pretty bad popular-science narratives.
First of all relativistic quantum field theory tells us that the vacuum is indeed the vacuum, i.e., it's really nothing. It doesn't move at all, it looks everywhere and at any time and in any direction the same. Also nothing fluctuates at all. It's just a state, where nothing at all is present (or more precisely all there is, is space and time or rather "spacetime" of special relativity).
Now, as soon as you put something in this empty spacetime its no longer the vacuum, because you put something there, and what's very important to remember is that whenever you want to observe something in fact you must put something in there, i.e., as soon as you want to observe the vacuum, you destroy it in the sense that you must put something you probe it with there (including also a measurement device, which is even a macroscopic object consisting of zillions of atoms and molecules).
Now quantum field theory is so complicated that we are not able to solve it exactly but we have some powerful math to solve it approximately in a formalism called "perturbation theory". All we can describe really exactly in this formalism are non-interacting particles, and that's why we are starting with them. Let's concentrate on electromagnetism and thus quantum electrodynamics, which describes charged particles and the electromagnetic field. Of course charged particles interact via the electromagnetic field and that's why we can observe the particles and the electromagnetic field. E.g., we perceive the electromagnetic field as light, because an electromagnetic wave hits the retina in our eyes and this leads to a signal due to the photoelectric effect.
But now, since we cannot solve the full complicated system of interacting charged particles and the electromagnetic field exactly we start with a fictitious system consisting of non-interacting charged particles and the free electromagnetic field and try to take into account the interactions in perturbative way. The idea behind this is that the strength of interactions between elementary particles like an electron is pretty weak. The coupling constant, quantizing this strength is the finestructure constant ##e^2/(4 \pi \epsilon_0 \hbar c) \simeq 1/137##, which is a pretty small value, and the hope is that we can describe the interacting particles and fields by calculating observable quantities as a power series in ##e## (or ##\alpha## for that matter).
In the 1950 Feynman found an ingenious way to depict these complicated pertrubations in terms of his famous Feynman diagrams, which look as if they would depict scattering processes, and to some extent they indeed do this, and that's why you can draw them also in pop-sci textbooks. The only problem is that they have to be understood in a quite delicate way, as particles and fields get a pretty specific meaning in the realm of quantum theory. What these diagrams really stand for are mathematical formulae which help you calculate all kinds of observable quantities like scattering cross sections, which tell you to which extent two particles (e.g., to electrons) are scattered in all possible ways allowed by the natural laws (conservation of charge, momentum, energy, angular momentum, etc.). That's what the formalism of quantum field theory provides to you and thus also the Feynman diagrams which are just a very clever notation for the corresponding formulas. So one possibility is that the electrons just get scattered on each other, so that after the scattering is over you again find two electrons. It can also happen that in addition to the electrons you also emit some radiation, and the quanta of the radiation are the photons. So you can emit one or more photons in addition to the electrons. Another possibility is that you create some photons and also some electron-positron pairs in addition and so on. What the Feynman diagrams tell you is the probability for each of these possilbe precesses to happen in a given way with all the particles in the final state with given momementa and energies.
Now there are also diagrams with loops. E.g., consider just one electron. Just a single electron line depicts a non-interacting electron. But now you can also have a contribution still having only electron lines as external lines but with some other lines building loops. These are called self-energy diagrams. In the same way you can also have a single photon line, describing a non-interacting photon, but also a diagram with still two external photon lines but with some loops in between. One is a loop consisting of an electron-positron line. These photon self-energy diagrams are also called vacuum polarization diagrams, because it seems as if indeed there were electron-positron pairs emitted and recombined back to a photon, but that's a wrong suggestion, because for reasons of energy and momentum conservation a single photon can never pop out an electron and a positron. Nevertheless the diagram with this loop is there and it's not zero. In fact it's even not making direct sense, because it leads to a diverging integral, but one has made sense of it with the socalled "renormalization", and such diagrams turn up also in other diagrams describing real particles, which are always depicted by the external legs.
One other important sort of observables are the spectral lines of atoms like the hydrogen atom, which consists of a proton and an electron bound together with the proton to form the hydrogen atom. Now you can calculate, again order by order in the coupling ##e##, the energy levels of the hydrogen atom. At lowest order you solve a wave equation called the Dirac equation. The result is already pretty accurate, but there are all the higher-order diagrams with loops giving corrections to these energy levels with more and more powers of ##e## or ##\alpha##, and indeed one can observe the tiny shifts of the energy levels corresponding to these correction terms. A famous correction has been measured in 1948 by Lamb, which is why it's known as the Lamb shift, and it triggered the theorists to think harder about the above mentioned divergent loop diagrams, which lead to renormalization (and a Nobel prize for Feynman, Schwinger, and Tomonaga developing this method). The amazing thing is that "subtracting all the infinities" in such a way that you can express everything in terms of the finite observables coupling constants, masses (and wave function norms) of the interacting theory (calculated approximately with the perturbation theory) and push the infinities of the loop diagrams to unobservable "bare" couplings and masses, corresponding to the unobservable fictitious non-interacting particles and photons, and then a kind of miracle indeed occurs: After all these complicated procedures just using a few experimentally found parameters like the charge and mass of the electron you find agreement between experiment and theory concerning the energy levels of the hydrogen atom at an amazing precision of several significant digits. It's among the best agreements between experiment and theory ever achieved in the history of physics.
