I'm looking for ways to make electricity magic appear believable in a science-fantasy setting. How would you go about explaining that a character in an otherwise science-abiding setting would manipulate electricity for combat, telekinesis, shielding himself from damage and maybe even teleportation. I'm looking for ways to give it a semblance of scientific grounding just like there have been attempts to link quantum entanglement to "the Force" as depicted in Star Wars. Coincidentally Force-wielders in Star Wars can learn all the skills mentioned.
These suggestions are fairly hand-wavy, but should provide some semblance of scientific basis for electric magic.
Two obvious areas to address are how the user generates the electricity and how the direct it.
Perhaps the user has organs similar to an electric eel's. You could attribute this to X-Men style mutation or genetic engineering.
Alternatively, perhaps the user has a large amount of nanites that work in unison to generate the charge.
Or, you could invoke quantum entanglement to set up a system where the charge is produced remotely and "teleported" to the user.
Using the electricity
Remember that you can't really push electricity through open air in a certain direction, you have to guide it by actually creating a path of least resistance to ground ("ground" in this case, hopefully being your target). The most basic option is by throwing wires or spraying liquid (electro Spiderman anyone?). That could easily explain throwing lightning bolts and such, though the user may have to employ some showmanship to get away with it.
To get a little more out there, you could come up with some way for your character to harness Bose-Einstein condensates. Some BECs are thought to superconductors, so if you could spray a jet of it at someone, you could use it to start off a lightning attack. https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate
You probably want to frame it as using the user's power to create an incredibly powerful magnetic field to redirect incoming attacks.
Magnetic fields could even be used to explain some forms of telekinesis, and not just Magneto style. Check out this video of a frog being levitated: https://www.youtube.com/watch?v=A1vyB-O5i6E
I'm not so sure there's a great way to make that plausible starting with electricity powers as your premise. However, you could have them able to move quickly over certain surfaces by generate a Lorentz force like a rail gun.
As a form of attacking, you could even create a weapon that fills the air with tiny particles (most likely metal) that your electric powers can use as a "path" to reach their target.