Yahoo has a news story up with a cool video showing a test firing of a railgun prototype.
http://news.yahoo.com/navy-railgun-tests-leading-ship-superweapon-2020-201003095.html
If the railgun only uses electromagnetism to launch the inert projectile, why is there so much smoke and fire exiting the barrel?
Is it some result of traveling at hypersonic (between Mach 5 and Mach 10, according to Wikipedia) speeds that is causing the flame and smoke?
Probably mostly from friction.
I read somewhere that there is electrical current passed between the rails via the projectile, so a lot of the smoke and flame comes from metal being vaporized due to the high currents and the resulting heat involved.
This didn’t come from anything even remotely close to an authoritative source though. Just some random guy posting on the internet. Whether or not he read it from a more reliable cite I have no idea.
Yes, that’s exactly how a railgun works - pass a current through a system of a conductor sitting on 2 conductive rails, and the conductor will experience a net force. It’s a simple beginning EM class problem. Make the current high enough to get the conductor up to Mach 10, and there are other effects.
Is it possibly a result of kinetic heating due to atmospheric friction?
If I had to guess I’d say it’s a combination of factors. First would be the air in the barrel that gets compressed from the moving projectile. Then there’s the enormous current passing through the rails and projectile which would ionize the air around them. It wouldn’t surprise me if what we’re actually looking at is plasma.
Nitpick: heat experienced by hypersonic objects is due to adiabatic compression of the air in front of the object, rather than viscous friction of the air against the sides of the object.
And in the present case, no, that’s not the source of all of the smoke and flame; note from the video (see at 0:32) that there is no smoke/flame emanating from the projectile once it leaves the barrel.
The Wiki page muldoonthief linked to calls it plasma, under the photo of a shot.
Muldoonthief is right.
Rail guns are quite commonly conflated with coil guns - the latter of which should not produce the dame sort of effects (but ate also less capable)
OP link: “Sorry but this video is no longer available”. 
Youtube has video at the moment: link
In my youtube link, there is a logo painted on the target with the latin(?) phrase “velocitas eradico”.
Speed kills?
Does this mean the rail gun must be repaired after every shot?
Possibly not with every shot, but probably at least with every few shots. This is still in the experimental/prototype stage of things. Once they get it fully functional with the range and power that they want, then they can work on making it survive more than a small number of shots before major maintenance is required.
I expect that there would be a modular design where the rails can quickly be swapped out. Similar to how you can swap out machine gun barrels when they overheat.
Given the Navy’s interest in it, I assume it will be placed on ships (eventually). Am I also right in assuming that the necessary electrical current will come from nuclear power? (i.e. electricity via a steam turbine?).
I couldn’t say if it’s the case here, but I know that in general railguns don’t “like” being fired from zero velocity–there is a high chance that the projectile will weld itself to the rails before it moves very far.
So I think it’s plausible that there is a relatively small conventional charge here to get the projectile moving, and the rails do the rest of work. One could probably use compressed air instead, but maybe that’s insufficient for their purposes.
This is getting better and better.
It could come from gas turbine generators as well. There isn’t any generator that could supply the necessary power instantaneously, so what’s done is to charge a large bank of capacitors that is then discharged when firing.
You could charge one up from an outlet in your house, but it may take a couple of weeks per shot. The generator requirements will depend on whatever the Navy specifies for recharge cycle time and the number of railguns on the ship.
As someone who works in the energy industry, I fully support this idea.