How do you extinguish a flares fire?

It's very difficult. Flares typically have a fuel/oxidizer mix contained within the flare, allowing them to burn even when not in the presence of oxygen. Smothering a flare will not work. You either need to let it burn out, remove enough heat so that the chain reaction ends, or remove the burning part from the rest of the flare. Police are trained to grind the flare in to the concrete until it is out.