How does Air Traffic Control know a flight's route and destination?
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)