It happens when something disrupts the flow of air through a jet engine. Often the cause is ingestion of foreign material--particularly a bird strike. (Certain mechanical problems, hail or ice, extremely heavy rain, turbulent airflow, and other things can also cause a compressor stall.)

Because the compressor isn't pulling as much air into the engine's combustion chamber, fuel doesn't get fully burned inside the engine. Hot unburnt fuel blown out the back of the engine ignites when it mixes with open air behind the aircraft, giving very spectacular bursts of flame. Crucially, what you're seeing is not the engine on fire, but rather fuel leaving the engine on fire.

Pilots are familiar with and trained for this issue, and handle it just like they did here: roll back the throttle until the problem goes away, attempt to re-apply power once stable airflow is reestablished through the engine, limit power or shut down the engine if necessary, and watch for further signs of trouble.