Tantrums are a stress response involving the "lizard" brain. The amygdala (emotions), and they hypothalamus (heart rate, temperature). Adults have similar responses, but our developed pre-frontal cortex helps us to regulate our reactions (most of the time).
What to do?
Don't try reasoning. The brain in a tantrum cannot reason.
Manage your emotions - step away (if possible) if you need to calm yourself. If you are calm, your child may mirror your emotional state.
Send out warm and empathic vibes - remember that your child's amygdala is screaming DANGER!! emphasize that he/she is safe.
Try some self-soothing strategies and tools BEFORE tantrums strike to see if your child can access these during the tantrum.
Remember, this too shall pass.