When a car backfires, it means that something has stopped working properly in the emission system in the engine. There are a number of things that can cause the improper fuel/air mixture which leaves unburned fuel in the exhaust manifold that may ignite. The sudden explosion of the fuel that was sitting is the cause of the loud sound that is generally referred to as backfiring.
Fuel filters remove various particulate, such as dirt and other contaminants that may be floating in the fuel before it enters the fuel system, and pumps the fuel to the engine from the tank. If the filter gets clogged that means the pump is damaged. When the pump is damaged, it means that there will be an incorrect fuel/air ratio, leading to the backfire.
Regulation of emissions in a car is handled by the catalytic converter. When this is working improperly or has been removed, chemicals are handled incorrectly and not converted to more benign chemicals before leaving the exhaust. This also leads to chemical buildup in the exhaust system due to the converter not disposing of the excess, causing the backfire.
The fuel/air ratio that is ignited depends greatly on the intake valve. If the intake valve is not opening and closing at the proper times, such as if the spark plugs fire before the intake valve closes, the spark will light the fuel that is in the intake manifold and cause the backfire.
