Scientists have puzzled over this question for more than a century. Although it is a simple question, it can be difficult to understand the answer. It is difficult to understand because humans love to connect similar things. Waves travel in water, for example, because water molecules accumulate and push out other water molecules as they attempt to move down under the force gravity. Sound waves also move in solids, liquids and gases because the atoms act like springs and oscillate when compressed from their equilibrium position. These examples work only because the medium is available for waves to travel through. You can’t have waves if you remove the medium.
Isaac Newton believed light was composed of particles. He proposed that light must travel through an invisible medium that causes it to scatter. Fresnel and Poisson were later to use this idea to explain the propagation of electromagnetic waves. These scientists believed the universe was filled by a stationary substance called “luminiferous-aether”, which allowed light to travel great distances away from us. They believed light traveled at a fixed speed relative the stationary aether. Since the earth is not stationary, they proposed that light travels through this aether and that the speed should be slower when measured in the direction that the earth is moving. The famous Michelson-Morley experiment, which used an interferometer for measuring the small differences in optical paths length depending on their direction, was conducted in 1887. To everyone’s surprise, the experiment showed no differences in optical path length regardless of how the interferometer was set up. Further experiments were to confirm that the expected effects from the earth moving through the aether weren’t present.
Albert Einstein finally published a 1905 paper that explained how many phenomena scientists had used aether to explain. He did this using ideas that he would later solidify in his work on special relativity. The debate over aether didn’t end immediately. As technology has improved, many more experiments have demonstrated with increasing precision that there is no phase shifting due to variations in optical path lengths.
I don’t believe there is a satisfactory explanation to explain how electromagnetic waves and light propagate in space-free environments. Although the mathematical concept of fields can explain how electrodynamic forces extend into empty space, it doesn’t really explain why. To understand electromagnetic fields, you need to learn electrodynamics. You can solve problems until you have a good understanding of how they work. It takes some time to accept the concept of invisible field lines that extend from charged objects and wrap around moving currents. Good luck. 🙂