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I understand that it isn't "fire" in the combustion of oxygen sense, but I can't find an answer online about what we're witnessing happening at the visible surface of the sun if all actual fusion explosive stuff is occurring near the center.
Would love a primer or a link to an explanation.
The "fire" on the surface arises because the core is really, really hot. If something has a hot interior, then its exterior will also get hot, even if not as hot as that interior is. You can see this in an everyday situation by thinking about an oven. If the oven has been running for a while, you will probably not want to touch the outside surface of the door, even if that is not the inside surface and not as hot. Heat transfers from the hot center to the exterior through the intervening matter in a number of ways, and the same is the case with a star like the Sun.
And at 15 million kelvins, that is a lot of heat transfer, so much so that the outside surface, even with on the order of $10^9\ \mathrm{Yg}$ of stellar matter piled on top, has to then be at 5800 K - over twice the heat of a tungsten bulb filament - in order to balance it. The reason that surface is cooler than the core is in part because it is a lot bigger than the core surface and also because the matter immediately surrounding the core transfers a sizeable amount of the heat right back in, thus slowing down the rate at which it goes out. Heat "percolates" from core to surface, instead of just rushing out all at once.
Insofar as the relation to fusion is concerned, the answer is that fusion keeps the Sun's core hot. It did not make it hot to begin with: that heat was originally provided by the gravitational contraction of the gas cloud that formed the star in the first place. That's how the core first got to 15 MK, at which point the rate of fusion became enough to sustain it there in an equilibrium by replenishing the heat being lost. (Actually, it was likely cooler. As the Sun has aged, it has steadily grown hotter, for reasons that are a bit complicated to get into in full detail but relate to the depletion of the hydrogen fusion fuel over time causing the star to contract and thus release more gravitational energy.) The surface would still be just as hot and interesting were there no fusion - although if that were the case, the Sun would gradually contract further and eventually become a dense white dwarf, and would have done so long before the present day.
Also, if you are asking about the various complicated phenomena on the surface, i.e. things like the "bubbly" appearance it has, the flares and prominences, sunspots, etc. - these things are all basically the result of the fact that all that hot gas also moves and churns thanks to the temperature gradient and, moreover, due to its heat, it is actually not just "gas" but a plasma, which then adds complicated electromagnetic effects into the mix as it generates its own magnetic fields with which it also interacts.
99% of nuclear fusion of hydrogen nuclei into heavy nuclei happens at the core of the Sun. This is because only at the core is the temperature high enough to overcome the coulomb barrier caused by the electrostatic repulsion between the nuclei. The solar core produces a lot of energy $(3.828×10^{26} W)$ by fusion and since all that energy has to go somewhere, it eventually reaches the surface making the surface of the Sun hot (6000K).
Most objects like a red hot iron or tungsten filament of a bulb glow when heated. An idealized version of this is called black body radiation. The colour at which the body glows or the frequency of the electromagnetic radiation depends on the temperature. Since the surface of the Sun is hot, it glows. This is not combustion but natural heat glow. Moreover glowing hot plasma rises up in columns while cooler denser gases settle at the surface of the Sun at speeds upto 7km/s in a process called solar granulation causing the Sun to look like a fierce fire ball.
Probably the "explosive" events at the surface that you are asking about are solar flares. The cause of solar flares is thought to be release of magnetic energy, because it is known that there are strong magnetic fields threading the surface of the Sun, and they get tangled up by the convection of the Sun as hot gas rises to the surface and cool gas drops back down. The gas density above the surface of the Sun drops very fast, so it is easy to heat that gas up to very high temperatures, though the details of how this hot "corona" is created are not entirely certain (it appears to be a combination of heating by pressure waves and various other plasma and magnetic waves). Usually that heating proceeds in small spurts, but occasionally a region of very strong and very twisted fields is created, and it can release energy more explosively, which produces a strong solar flare and often comes with a "coronal mass ejection."
So the bottom line is, the physics of explosions at and above the surface are magnetic in nature, and are very different from nuclear fusion. These "explosions" are only possible because the density is so low, which is also why fusion is not important, even though the temperatures are comparable.
The fusion reactions only happen in the central regions, because immense heat and pressure are essential for these reactions to take place. The mutual repulsion of like charges has to be overcome before the various nuclei can fuse. Then the heat and energy generated has to find a way out, otherwise the star would explode, so this heat and energy gradually makes its way to the surface, taking some time to do it. Fortunately some energy is instantly carried away by neutrinos, which reach the surface at the speed of light. However, scientists are baffled by the surprisingly high temperature of the corona, which mysteriously is much higher than at the sun's surface. So far, no one has come up with a mechanism which explains this.
