Awesome question! To answer it, we need to understand what light is. First, let's start with electricity and magnetism: it turns out that if you have a magnetic field and wiggle it (that is, you make it change in time), you can produce an electric field. For example, if you take a magnet and move it through a metal ring, the changing magnetic field (from moving the magnet) will create a current in the ring (you can try something similar to this at home if you have a magnet and a narrow metal tube: if you drop the magnet through the tube, it will fall very slowly due to currents being created in the metal). Similarly, it turns out that if you have an electric field and wiggle that, you can create a magnetic field.

So what does this mean? Well, say you start with a varying electric field; then that'll produce a magnetic field. But if that magnetic field is varying, then it'll produce an electric field, which will in turn produce a magnetic field, and so on. So in this way, you can create a self-sustaining wave of varying electric and magnetic fields: an electromagnetic wave.

So, where does light come into this? Well, we just described what light is: it's an electromagnetic wave! And because the electric and magnetic fields need to vary to generate one another, the wave has to move; it can't just sit still.

Light travels in tiny packets called photons. These packets aren't actually particles: they don't have a mass. The exhibit may properties of waves (diffraction for example, like why a shadow is dark but not pitch black) but aren't really waves either, since they can be "quantized." Quantized means they can be counted, essentially, whereas waves cannot necessarily be counted. This is one of the ideas of quantum mechanics (quantizing things). So, light moves because the photon has wave-particle duality, in that it acts like both a particle and a wave. Electromagnetic waves move... because they are waves. That sounds like a strange answer, but think of ocean waves. You can't truly lock an ocean wave in place and have it still be a wave, traveling is inherent to what waves are. In short, light travels because it behaves partly like a wave.

Light carries energy but has no rest mass, so it has to move. It's a wave of electric and magnetic fields, alternating with each-other.