Why is my battery light icon coming on?

You have a charging problem. It's usually the alternator in cases like this. To check this, have the car running on it's own (after jumpimg remove the jumper cables) then measure the voltage at the battery, the voltage should be between 13 and 14.5 volts or slightly higher. 13 volts is already a bit low and maybe a sign of bad diodes in the alternator. Replace the alternator if the voltage is below 13 volts. There is a chance that the field sense wire going to the computer is dirty but I highly doubt that. Most times it's the alternator. Personally I would replace the alternator if the voltage is below 13.5 volts. Another way to check this is to have the car running then disconnect the negative (black wire) battery cable the car should remain running, if it dies out replace the alternator. If it stays running then turn the headlights and accessories on the alternator should be able to handle these things without the battery, if it dies out replace the alternator.