You're absolutely correct, fear of flying is often a manifestation of some underlying psychological issues. Mostly agoraphobia and claustrophobia. Which seem at odds but not really. General anxiety disorders play a part as well. So hence the reason why I always recommend seeing professional guidance, such as fear of flying clinics etc.
I also recommend to people who are mildly anxious to get familiar with how to get to the airport and the terminal etc. Familiarity reduces the stress.
How do aeroplanes fly?
Serious answer. Benoulli and Newton. Physics mainly.
Real answer. Money. No money, no aeroplane will fly.