In the real world, the “vaporise” setting in SciFi ray-guns comes from a desire to make extras disappear quickly when their characters are killed off.
As countless of pedants have noticed, a real-life weapon which vaporised a target would have all sorts of unpleasant side-effects, from the merely icky of inhaling your enemies to the potentially fatal of suddenly adding 80 cubic-meter-STP of (lethally hot) gas to your room in less than a second. There are also, shall we say, artistically convenient behaviours such as one scene in Star Trek VI where a pan is vaporised while the mashed potato inside is untouched.
None of these things ought to spoil your experience of Star Trek or similar — they’re character-driven space soaps, not hard SciFi. That doesn’t mean it’s not fun to invent some plausible-sounding rationale for how it fits into the in-universe technobabble.
So, phasers, with a digression via subspace.
In the Star Trek universe, “subspace” is treated as an extra dimension — not up-down, not left-right, not front-back, not future-past. If you’re “deep” in subspace, you’re “further” from the reality we know about. How this works precisely is never described, so let’s pretend that there are a bunch of different stable layers that matter can occupy, one for normal space and one more for each different integer warp speed; and lets assume that in the absence of anything pushing you between layers, you just stay on your current layer.
What if the “rapid nadion effect” is a nudge in the subspace-realspace direction?
All the atoms in your body are connected by relatively strong interatomic forces, while the interatomic forces between different objects are much weaker (not zero, but much much weaker). Let’s say you’re hit by a beam which nudges you in the direction of subspace: if its a weak push, the atoms hit by the beam are briefly a little bit outside normal space, but they rapidly return. This effect propagates through your body in exactly the same way that a sound wave would — each out-of-place atom drags nearby atoms with it, but they’re quickly restored to their original place. This could stun you or kill you, depending on how much it interferes with the chemistry that keeps you alive, in much the same way that a punch or a grenade both send waves through your body yet have very different impacts on your life expectancy.
If this nudge is strong enough to push your body to the next subspace layer, the part of your body first hit by the beam will seem to disappear entirely, without the inconvenience of exploding! All the atoms bound to the nudged-into-subspace patch of flesh will be dragged with it onto the next subspace layer, which does not have any air. If this happens at the speed of sound in water (5336 km/h), a 2m humanoid hit in the middle would disappear completely in about 0.7 ms. The effect you see on screen is a far more prosaic 8-15 km/h — again, don’t worry to much about that: Trek has very little in the way of scale or time consistency, but even if it did you should pretend it’s a dramatic slow-mo.
There’s no narrative requirement for subspace layers to be limited to three spacial dimensions, so we can also posit that subspace is (e.g.) 4 spacial dimensions. In 4D, a creature like us built in 3D space would fall to pieces in much the same way as you might expect a creature built out of a single layer of atoms sandwiched between two plates would almost instantly disintegrate if you took the plates away. One idea which would allow “being pushed into subspace” to be much more dangerous than “being in a starship when the warp drive is switched on” would be another fairly ambiguous piece of Star Trek tech: the inertial dampeners. The inertial dampeners are supposed to be space-filling forcefields which push every atom in your body at the same rate the ship accelerates so that you don’t feel any G-forces — vitally important when you go from zero to 0.25c (74,770 km/s, full impulse) in one second. Those very same force fields could (with enough technobabble) keep the crew from disintegrating even if they were in a 4D (or 5D or 6D or…) subspace domain.