Recently, I have stumbled upon a YouTube video by Veritasium describing the conductivity of fire. My question is: how exactly does fire conduct electricity? I am a high school student; therefore appropriate language is expected.
This video illustrates how a candle flame conducts a high voltage (10,000 volts). Although the ion density is small in a candle flame, they are sufficiently present to conduct electricity.
It basically conducts electricity the same way salty water does: both contain some concentration of charged particles that are free to move.
Water contains some concentrations of ions and protons (H$^+$ protons). When there is a voltage difference, the ions will move according to their charge.
The hot gas of the flame contains positively charged ions and electrons, which will move in the same way as the ions in the water. The reason there are ions is that the heat of the gas is such that some of the electrons can free themselves from the attraction of their atoms.
At the (relatively) low temperatures of a candle flame, not many electrons are free, but with sufficiently high voltage, you can get a good current. At much higher temperatures, there are so many free electrons and positive ions that physicists say the matter is in a 4th state: plasma. Plasma is found in stars, solar wind, fission reactors.