If you set the fee to 0, the payment will still reach the other party, it might only take days instead of minutes. correct?
Correct - but not 100%
if you will put fee 0 and transaction will be bigger than for example 0.001 it will take few hours, not days...
Anyway transaction will be confirmed then
so you don't need to pay - it is just optional and make transaction muuuuch faster and miners more happy!