What happens when a takeoff clearance is cancelled by the ATC?
What happens in the control room?


Aborting a take-off at high speed is dangerous, even if the aircraft is travelling below V1. If the aircraft is already rolling, the controller will make a quick decision whether the situation justifies an aborted take-off. For example, if there is traffic infringing the runway and a crash is imminent the controller would attempt to stop the aircraft. If an aircraft is attempting to take-off from a taxiway in snowy conditions it might be safer to just let him take-off unless there is other traffic on the taxiway. 

If the aircraft hasn't started rolling the decision is of course easily made. 


  What does the ATC do and how does he/she make sure pilots understand
  the cancellation?


Using standard phraseology is the best way of making sure the message comes across, especially in international aviation. 

If the aircraft is not yet rolling, the proper ICAO phraseology is:


  Tower: Big Jet 345 hold position, Cancel take
  - off, I say again cancel take
  - off (due to vehicle on the runway)
  
  Big Jet: Holding, Big Jet 345


If the aircraft already started rolling the controller would say:


  Tower: Big Jet 345 stop immediately, (Big Jet 345 stop immediately )!
  
  Big Jet: Stopping, Big Jet 345


No immediate verbal reply is to be expected as the flight crew have other things to care about at that moment. The effect of their actions should be clearly visible from the tower.




  How do the pilots in the aircraft taking off respond? Are they trained for this scenario?


The crew will assess whether they can stop safely. Below V1, they will probably abort the take-off, above probably not. The final judgement is made by the captain. He might reason that it is safer to get airborne and pass the infringing traffic overhead than running into it at full brakes. They are trained for aborted take-offs, but since there are so many non-standard situations the final decision is made on the basis of a mix of training and expert judgement. 


  Is a takeoff cancelled even after V1?


Since ATC does not know when the aircraft passes V1, a controller might attempt to cancel the takeoff after V1. The flight crew will only attempt to cancel the take-off after V1 if it is clear that there is no way the aircraft can become - and stay - airborne safely. In such a case it becomes damage control rather than damage prevention.