Aerobars are well known to the triathlon community due to the no-draft nature of most races. (Not ITU.)

Aerobars are the single biggest speed-gaining technology you can give your bike, far exceeding $2000 race wheels, $3k aero frame, etc. A $80 pair of aerobars clipped onto a road bike with a good position will give you 3-4x the speed savings, if not more, of those race wheels+aero frame+aero helmet. As said, the aerobars decrease your body's aerodynamic penalty - and since the aeroness is 80% rider and <20% bike, the effect is much greater than making changes to bike alone.

Speed gained will be about 1-1.5mph. Doesn't sound like a lot, but remember that this speed gain often happens at the top end of your speed, and since the amount of power required increases exponentially for increases at higher speeds (meaning you'll require a LOT more power to go from 22->23mph as compared to 12->13mph), the difference is even huger than you might think.

It won't suddenly make you a lot faster, and without a speedo, you'll wonder if you're actually getting any gains from them at all since it's pretty hard to pick up 1mph subjectively. But you'll definitely notice when you're keeping track of your speed and really trying to hold a high steady speed in a TT - it'll surprise you when you're at your 'redline' speed and still going strong.

Don't expect to catch anyone blowing by you before, though. A 1mph difference works out to a very slow pass, one that you should be able to 'catch' for a good while without aerobars. If someone unceremoniously dumps you quickly, there's no chance that you'll catch them just with aerobars.