"In the jargon of field theory, physicists often say that “virtual particles” can briefly and spontaneously appear from the vacuum and then disappear again, even when no one has put enough energy into the field to create a real particle. But what they really mean is that the vacuum itself has random and indelible fluctuations, and sometimes their influence can be felt by the way they kick around real particles."
I can't help but immediately question every jargon word I see, especially "random", "particle" and "wave".
Im my experience, all terms in physics (like "particle", "wave", "energy") are highly context dependent and most PhDs could spend hours debating what is actually meant in a given case. Such discussions almost never lead to publishable results and are thus considered a waste of time, or leasure at best.
Usually, you just "shut up and calculate". Meanwhile, the calculations are motivated by "intuition", which involves combining known or unknown reasonable approximations with a basic theory. This process is never explained systematically and rather the hope is that it will be absorbed via osmosis by the brighter students.
It turns out, you actually don't need to have a coherent concept of "what a particle is" to perform particle physics experiments and evaluate the data. Sometimes, when pressed, operational definitions can be offered. For example, I've heard a professor say: "a particle is defined as a bump at a given energy in this plot". I'm not sure how ironic that was supposed to sound...
Think its the other way around really.
Decades of contradictory analogies and false intuitions later and "shut up and calculate" is still the best way to understand it.
A field's normal vibration is like plucking a guitar string. A quantum is like shaking the end of a rope and seeing a ripple fly down it.
Solitons are self re-enforcing wave packets in some medium, where the dispersive effects get canceled out. I like to think of field quanta as being similar at a very high handwavy level. Of course the trick with quanta is only certain energy levels are allowed.
The peculiar experimental challenge about the Higgs field is that it can extract its quanta from certain physical states (certain initial conditions in a particle physics reaction) only at very high energy and with low probability, but that is true also for other particles. Its truly peculiar thing is that the presence of the Higgs field, in addition to the fields of all other particles that we know of, explains why quanta in general have a mass (although this is not clear for neutrinos).