In France it works pretty well because the government paid for it, and subsidizes it, but flights often are cheaper than the train. The main thing about the TGV is it can drop you in the middle of whatever city while the airport may be well outside of it.
My understanding is that in most of Europe the subsidy is either directly or indirectly paid with fuel taxes that make auto and air transport more expensive. Thus the subsidy and tax act together to both push and pull travelers to the trains.
Note that this isn't the only reason trains are so much more practical in Europe. The biggest reason is that Europe has substantially higher population density which obviously makes public transit much more practical.
Building off of Beta's LA->SF example:
Per google the flight (LAX to SFO) takes 1:10 with prices starting at $108. Per Google Flights the price ranges from $108 to $359 but I would guess that most travelers are paying around $150-200 so call it $175. The time ranges from 1:10 to 1:36 so call it 1:25.
To make the same trip by car Google says it is 379 mi and takes 6:20. 379 mi at 30 MPG would cost you 12.6 gallons of gas. According to Google, gas in California is about $5/gallon so that is about $65 in gas to make the trip by car but gets cheaper per-passenger as you add people unlike flying which costs the same per traveler.
Per Beta's example the train would take around 2:40. Lets say, just for sake of discussion that a "free market price" for the train would be about 75% of flying so about $130.
Now for someone traveling alone the options are:
- 1:25 to fly, costs $175
- 2:40 by train, costs $130
- 6:20 to drive, costs $65
It is a pretty simple analysis at that point, if you start from flying:
- Train costs you 1:15 and saves you $45. That is $36/hour.
- Driving costs you 4:55 and saves you $110. That is about $22/hour.
Most of us are going to pay the $36/hour to upgrade from the train to the plane and nearly all of us will forgo driving. The main exception would be if someone is taking their family. If you have four people on the trip the numbers change dramatically:
- 1:25 to fly, costs $700.
- 2:40 by train, costs $520.
- 6:20 to drive, costs $65.
Starting once again from flying:
- Train costs you 1:15 and saves $180 (same math as above just x4).
- Driving costs you 4:55 and saves you $635. You might actually spend the time in the car if it saves you $635 depending on what you are doing and how big of a hurry you are in.
Now if we institute a tax that increases the price of flying or driving by 30% while decreasing the price of the train by 30% we get:
- 1:25 to fly, costs $227.50
- 2:40 by train, costs $91.
- 6:20 to drive, costs $84.50.
Or keep going. Eventually the train becomes a viable option for enough people to make it work.
