thank you ZapperZ and jtbell for helping to clarify this issue.
I think the original question being posed was about simple relativistic collisions and not the Higgs mechanism, to start off. So let's begin with E=mc^2
This is a relationship between mass and energy. This means, in one way of phrasing it, that mass and energy are equavelent. Mass is a form of energy, just like kinetic energy is a form of energy or classical potential energy (at least at this level of presentation). Notice that E=mc^2 is an equation where m describes the total (or relativistic) mass of the particle. E is then the total (relativistic) energy. The minimum total energy that a particle (or system of particles) can have is its rest-mass energy, E0=m0c^2--m0 is the rest mass, or the mass measured when the particle is seen to be stationary. This means that a mass isolated from all forces and observed to be stationary will still have energy, no matter what--it is energy. But once that thing is moving, it will carry kinetic energy, as we well know. The total energy will be then be KE+E0 (in the nonrelativistic limit; relativistically E^2=(pc)^2+(mc^2)^2 ) , or the kinetic energy plus the rest mass energy. In classical (nonrelativistic) interactions, KE<<E0, and because no new particles are being created or whatever, E0 is ignored when taking into account energy exchange (you can take a Taylor expansion approximation resulting in the classical energy expression from the relativistic). Note that I am not including effects from potential energy fields, as this involves more complicated theories and mathematical constructions.
Ok, but now realize that if KE+E0=E=total mechanical energy, at least roughly speaking in the classical limit, E=mc^2 still. This means that as KE increases, m increases! So, if you really do have an extremely hot fluid (plasma), the mass of this ensemble would be significnatly increased. But then of course the temperatures are ridiculously high and you would feel these high-energy particles burning into you and your equipment, not being able to measure its mass by conventional methods. My understanding is that, in ordinary chemical reactions, the change in mass due to bond energy released from broken checmical bonds from high temperatures does alter the mass you measure, but this depends on the properties of the material and how precisely you are measuring.
All right. Now this applies to particles with mass. A photon does not have mass and so E is obviously not mc^2 because its energy is not zero. If the energy of light were zero you wouldn't have solar energy, photosynthesis, etc. E=hf according to Einstein, and this determines the energy of a photon.
So now the question is, how do relativistic collisions generate 'mass' from 'energy.' Though the specific particles I mention are not of consequence to the principle illustrated here, consider an electron and a positron (anti-electron) that are both moving very quickly relative to each other. Let us choose a center of momentum frame so that the momentum of the electron and the momentum of the positron are equal and opposite (analogous to the center-of-mass frame in Newtonian mechanics). If they are both moving at extremely high velocities they will have very large kinetic energy. But remember a large kinetic energy means a large relativistic ('total') energy, and thus a large relativistic mass. So now the two particles collide. Because of the rules of relativistic quantum mechanics (and more generally particle physics), if the sum of the relativistic energy of the two particles is large enough, it can create two more particles that are more massive (i.e. their rest masses are greater), provided that the relativistic energy is conserved. What will happen is that the resulting particles will have a lower momentum because the rest mass energy is larger than it was before the collision; E0 increases, so KE must decrease to maintain ~E0+KE=E (correctly: E^2=(pc)^2+(mc^2)^2 ). If you think in terms of relativistic mass, in fact there was really no mass generated at all during this process, because the total energy stayed the same and hence the total relativistic mass stayed the same (the two are proportional by c^2).
When you get to the quantum field theory of it, the pair annihilates to form a photon which then decays into other suitable particles, such as a muon anti-muon pair (chosen particularly to suit conditions in particle physics that I don't want to discuss). Note here that the photon is massless, which is not the case for more complicated interactions (with massive exchange particles). Then, the question is, how does the creation of a massless virtual photon change two particles with mass to something without mass, and then into two things with mass again? Recognize first of all that when these things "collide" or interact, they are on distance and time scales that are very small and due to the Heisenberg Uncertainty relation this a corresponding allowed increase in the uncertainty in the energy and momentum of the exchange particle, which really is why the exchange particle is 'virtual' in the first place. Note though, that no matter what, on large enough time scales for the Heisenberg Uncertainty Principle to be noncrucial, there is no violation of the conservation of energy, and hence there is no mass being 'destroyed' because energy and mass are equivalent. This is the main point.
Now if you want to go deeper and say, 'What do you mean, the particle pair generates a virtual photon that then sponataneously generates another particle pair with different rest mass?' This is a tough question. The way that quantum electrodynamics works as formulated today, it is a perturbative theory. When the electron and positron collide we don't know exactly what will come out. It's not like photosynthesis, where you just add light energy and you get what you want to get. There are certain probability amplitudes telling you what you are likely to get depending on how much energy you have. If you don't have enough relativistic energy to make a proton and an antiproton, you're just not going to make them (this is why we need ever-larger colliders to make ever-bigger particles). But if you do have enough energy, there's no telling whether you're going to make a proton & antiproton, muon & antimuon, quark & antiquark, etc. There are certain probabilities for each depending on various factors, but we cannot deterministically calculate what will result.
On a more fundamental level we do not even know what really 'causes' this 'generation' of mass; we do not know exactly what happens to the mass when matter and antimatter annihilate in the sense that we do not know about a specific mechanism we can use to predict and calculate what results from such 'generation' or 'destruction' of mass. But again remember, mass and energy are equivalent and energy is not destroyed. Truly, the conservation of energy-momentum is the most fundamental conservation law we have, and the day we lose that is the day we lose everything know about the way nature works.
So I think the answer to the original question was that we aren't just creating mass out of nowhere, there are certain conservation rules that must be respected, and these rules are what make creating a suitable mass-generation theory difficult and elusive.
