There are several reasons why flying can be so cheap in Europe:
1. The discount airlines have become masters at turnaround times. When the planes are sitting at the gate, they're not making money. They slashed these as much as they could.
2. They only fly one model of airplane (typically either the 373 or a320) to simplify maintenance.
3. Most major airports in Europe are privatised. One of the effects of this is that they've become giant malls that want to get people through them, so landing fees have steadily gone down (there are edge cases where the airports pay the airlines to go to them, but those are rather rare exceptions).
4. Some airlines (more specifically Ryanair) fly to cheaper/more outlier airports. If you take a Ryanair flight to Paris, you're landing in Beauvais Airport, which is almost halfway to the english channel. You'll then have to take a bus. When in europe, I usually prefer Easyjet for the reason that though they're more expensive, you're landing at an airport with decent transit to the city you're visiting.
5. There's no frills at all. If you're just flying with no luggage, you won't believe how cheap it can be (I flew from madrid to london for 25 euros). But the fills aren't cheap. Checked baggage is more than the cost of the flight, food is a rip off, etc.
Also, trains in europe are still often state-owned (not the case everywhere, I know), so they're forced to run uneconomical lines for political purposes, etc.