Earlier quoted context omitted.
So long as fuel is reasonably priced, planes can take more people farther, faster it is simple economics. Trains are still preferable for local trips and even some medium range trips, but to go all the way across Europe by train would be terribly expensive compared to taking a cheap flight.
That doesn't really answer the question. What about planes is cheaper? For simple economics to apply, they have to do something cheaper than trains, and it's not obvious what that is. As far as I know, for example, trains are significantly more fuel efficient per passenger-mile.
Your statement that "As far as I know, for example, trains are significantly more fuel efficient per passenger-mile" definitely has some merit, but... trains are only more fuel efficient per passenger if they are as full as planes. An Airbus with 180 passengers might beat a train in per passenger miles if the train is not loaded.
Considering that, trains definitely are cheaper for highly utilized routes (commuting routes, for example), but for a leisurely trip across Europe, an Airbus packed to the gils could be cheaper.