Your quote suggests that AM-18 aborted take-off because of ATC instruction, but if you read the article carefully it's not the case.
The AM-18 was accelerating for take-off when the crew noticed something is wrong with the tires (probably blown tire already) and aborted just below V1 due to their own decision.
- Exactly what happens in the control room?
I don't know exactly, but there are various sensors designed to warn of possible runway incursions. Apparently the sensors detect aborted take-off and alerted the controller.
In some places, including USA, the controllers will clear aircraft to take-off or land when they expect the runway will be clear by the time the aircraft reaches it. So they have to cancel that clearance when there is an incident and the previous aircraft does not vacate the runway as happened here. In other places, e.g. Europe, they don't do this, so they don't have to cancel any clearances, but the pilots are often receiving their clearances at the last moment which has it's own disadvantages.
- What does the ATC do and how does he/she make sure pilots understand the cancellation?
They cancelled clearance of the next flight that was just about to line up behind AM-18. I believe "cancel" is standard wording, so something like "tower, XYZ, take-off clearance cancelled; line up and hold." They know the pilot understood since the pilot reads back the instructions in reply.
If the aircraft has already commenced take off roll (which didn't happen here), time becomes critical, so then the instruction is just "XYZ, stop immediately", often repeated twice, or until the aircraft starts to brake or the pilots answer.
It does happen now and then. See for example this incident.
- How does the pilots respond? Are they trained for this scenario?
If the clearance is changed, the pilots simply follow the new instructions.
If "stop immediately" is requested, pilots abort the take off if possible.
If the pilot does see the problem (nobody will request aircraft in take-off roll to stop unless there is a serious risk of collision), which they often do, they may decide to continue the take-off if it's more likely they'll lift off before the obstacle than that they stop before it. Captain is the final authority responsible for safety of the aircraft, so it's their call.
- Is a takeoff cancelled even after V1?
No, because the aircraft is no longer able to stop on the remaining runway. Instead the pilot may rotate earlier. The Vr is higher than Vmu, so they can rotate at slightly lower speed and they can increase thrust if they are doing reduced thrust ("flex") take-off, which they now usually do.
For example in this incident ATC instructed the crew to abort (but failed to follow correct phraseology). At the same time the crew noticed the problem themselves, but they were already above V1 and decided they can fly over the obstacle and not stop before it and therefore continued the take-off.
Note that at V1 the aircraft is already long way down the runway, so the obstacle is likely near the end of the runway and likely can be already seen by the pilot, so the pilot decides how they try to avoid the collision.