How does Air Traffic Control know a flight's route and destination? I am trying to find out in layman's terms how does ATC determine the direction a flight has to take after takeoff, or when should it start descending for the destination airport. I understand it has to do with the flight plan but Googling revealed lot of technical stuff that I couldn't understand. In my limited experience of listening to LiveATC and YouTube video, ATC never inquired about the destination - it felt like they already knew. 

Is destination information always available on the screen to ATC or does the flight crew request changes based on their flight plan? How does it work when you cross international boundaries? Is the same flight plan available to ATC in all the countries the flight will be flying through?

For example, a flight taking off from SFO could be northbound towards Seattle or southbound to LA or eastbound. ATC has to make room for the proper direction of the flight and hence needs to know the destination.

To complicate matters, I have seen flights taking different routes while the destination is same e.g. one SFO-FRA flight hugged the west coast (magnificent views of the coastline) up to Canada before making a right turn, whereas other times - but with the same flight number - it kind of crosses the continental United States diagonally. So I feel it's not just about the destination, there are other factors involved.
Under instrument flight rules (which all scheduled passenger flights operate under), the pilot will file a flight plan with the local air traffic control authority (in the US, that is the FAA). This flight plan describes in fairly specific terms where the pilot intends to fly, by going from point to point. You can see flight plans on a site like https://flightaware.com.

These flight plans are entered (electronically, in most cases, but sometimes typed in) into the air traffic control system. So each controller who handles the flight knows where it is supposed to go next.

The details of the start and end of the flight isn't determined by the pilot, but by ATC when they set up the departure or arrival. These are done using instructions called SID (Standard Instrument Departure) and STAR (Standard Terminal Arrival Route). Depending on the traffic patterns around each airport, ATC may assign a particular SID or STAR. These instructions are like "fly at this altitude until this point, turn right, descend to this other altitude, fly until you get to this other point", and so on.

For international flights, the flight plans are transmitted from one country to another using AFTN (Aeronautical Fixed Telecommunications Network). So each country that the flight flies over knows where it is supposed to be going.

Other factors which can influence the actual route a flight takes can include:


Weather
Other air traffic
Military operations
International agreements (or more specifically, disagreements)