So, how would skeletons be able to move?
How would magic hold the bones together and make them able to move?
How would they show gestures and expressions if they don't have muscles or nerves?
Would magic be an element that has a unique type of atoms that allows skeletons to move? How exactly?
(I know I'm probably overthinking the logic of this by the way!)
Temporal Displacement of Inertia
The bones aren't actually connected by anything. Rather the magic that animated them temporally displaces inertia from the past to move them in the present.
When you're practicing your sword fighting and after a while the sword seems really heavy ... it's not that you're tired, it's that sometime after you die you're going to be reanimated as a skeleton.
Magic has its own insane logic, like how you can use a lock of someone's hair to curse them but you can't use their fingernail clippings, and hair cut by a silver blade is better than hair cut by an iron blade, and a blade forged under moonlight is better than a blade forged in the daylight. For every "why" and "how" there's an answer but the answers only raise more questions, magic is fundamentally illogical but it's predictably illogical which makes no sense except that it does because it's magic.
You could animate clay but skeletons are better because they remember being a person, that person may have been your enemy in life but as a skeleton animated by you it will, at your command, murder its former family without remorse and perceive no incongruity in its behavior. It's a skeleton animated by magic, not a person, it acts like a person, thinks like a person, remembers being a person, but fundamentally it's not a person it's a magically animated skeleton that only wants to carry out your will.
A skeleton walks around on bare bone feet, it's every joint creaking (unless you tell it to grease itself), it's bones grinding away and decaying but it doesn't care, there's no muscles or tendons but it still moves as though it has them, it'll sit around with other skeletons "drinking" from empty bottles and stumble around drunkenly afterwards, it'll complain if told to hide underwater, it doesn't need to breathe but still objects the indignity of being treated so inhumanely.
I imagine that the original owner of the skeleton would need to be summoned in some way. It takes long enough for humans to learn to control their own bodies that you wouldn't want your magic user to have to learn all the details of how to realistically control large numbers of other people's bodies. The spirit would have the knowledge of how to move their skeleton even without the muscles being there, kind of like how amputees can still mentally "feel" and "move" their missing limbs even without the muscles and nerves there. The magic would give the summoned spirit the power to actually translate that knowledge into movement, and would give your magic user the power to control the summoned spirit.
Gestures and expressions shouldn't be too difficult. A lot can be communicated through posture, sound, and motion - think of BB-8 or R2-D2 from Star Wars. Neither of them have a face or even a humanoid shape, but we as an audience still manage to understand what they are "feeling" and their reactions to things.
The question is not "how?", it's "why?". The answer to "how" is "magic". Magic does what magic does... magically.
And magic could create invisible forces/entities that move and strike and lift and crush. But magicians have a sense of style and decided to embed bones in their invisible entities to add that flair... that palpable fear that distinguishes a magician from a technician.
Sort of like painting a face on the front of an A-10 attack craft. All of the technology in the plane would still work to rain destruction down upon its targets. But depleted uranium pellets spewing from a monstrous mouth on the front? Priceless.
(So the indirect, mundane answer is that magic is not animating the skeletons, the skeletons are embedded in an otherwise invisible, autonomous, pure-force weapon system.)
Telekinesis would allow for the body to move, but a "mind" would have to be in place to control it. If the spirit or some other essence of the skeleton existed, it could develop telekinesis to tell its bones to move, but only be able to move it in ways it expects it to in the first place (kind of a psychosomatic thing, it works because it expects it to be able to work).
It's magic really, but magic you could do science around. Plus, all the movement would be completely natural since the skeleton is literally just moving as the dead person's will expects it to. After it first moves, it grows confidence that it can and so never loses that ability, so it just needs to move once by mistake or because it doesn't realize it's dead yet.
The most powerful of them would likely be the ones with the most confidence in their ability to move, so the ones that were stronger in real life, and all of a sudden you basically have skeletons that are just like their humans but dead.
This could also be a funny explanation of some people not knowing their own strength or being much stronger than they look - they're actually moving with confidence which allows their powers to give them strength. Very flowery.
TL;DR: A group of people could have telekinetic powers but not know it until after they die. The part of them that's capable of thinking lives metaphysically so it can perform subconscious telekinesis on the skeleton by expecting to be able to move because it doesn't realize it's dead, and then be able to move because it knows it can.
Just to throw this in there, maybe when you activate the magic, it creates tendrils of magic energy or use existing materials like vines (think druidic or necromantic magic) to support and make the bones move like a puppet on strings.
