How do aircrafts cross the skies without hitting each other?

Aircraft fly according to ATC clearances provided by Air traffic controllers.  Air traffic controllers apply surveillance (i.e radar or ADS-B) or procedural separation standards to keep the aircraft apart.

To avoid conflict situations and collision between aircraft, ATCOs use several seperation rules based on concerned aircraft positions. Vertical or lateral seperations are applied based on positions defined from Ground NAVAIDS, Surveillance systems such radars and ADS-B.

aircrafts follow which path to take based on the flight plan and the air traffic clearance given to them. Pilots continuously in contact with ground control stations and air traffic control through radio base communications and radar

aircrafts are having a continous ground control stations guidance with previus flight plans wich have all the required information including hights   

air traffic controllers services and use of ATM/CNS

By utilization of air traffic procedures including issuance of clearances, implementation of traffic separation such as vertical, longitudinal and horizontal.