Well, my guess is that the best way to explain the effect (I’m assuming the effect is real) is by noting that a ‘fire’ can be thought of as two things: a chemical reaction where a fuel is being oxidized, and flames, which are hot gases and soot rising off of the fuel. The flames carry heat away from the fuel, and often onto other fuel, extending the fire. The flames also allow fresh air to flow in underneath, providing more oxygen for the reaction.
If the soot particles are indeed charged and can be pulled by an electric field, they’d tend to draw the gases with them. So you could control where the flame goes so instead of rising straight up as a normal flame does, it would point sideways to some degree.
Now, I don’t know how much you really could do – I suspect the researcher is saying “maybe you could do these cool things or maybe the effect is too small to do anything useful, but we don’t know yet. Let’s find out.” But if the effect ends up being big, then you could pull flames sideways (again, by pulling the soot particles in the flame) or even try and get them to sit back down on the fuel. If you got the flame to sit back down on the fuel, not much oxygen could get in and the fire would go out (just like a match in zero-g and normal atmosphere will go out by itself). Getting a flame to move downwards seems pretty tough though, so I don’t have huge hope for an electrical fire extinguisher.