Who makes sure that planes fly in and out of airports safely and do not run into each other in the skies?

The correct answer to this is:

"Air traffic controllers"

These people are the personnel who is responsible for the safety, orderly, and swift flow of planes flying in and out of airports using the global air traffic control system. They are generally stationed in air traffic control centers and control towers on the airport ground. They monitor the position, speed, and elevation of aircraft in their designated airspace visually and by radar, and give directions to the pilots by radio.