Intergalactic space is estimated to have a mean density of about $1$ molecule per cubic meter.
Air has a density of about $ 3 \times 10^{25}$ molecules per cubic meter.
1 Light Year is about $9 \times 10^{15}$ meters.
A crude bit of multiplication would thus suggest that a photon passing through 13.5 billion light years of intergalactic space has about as many encounters with molecules as a photon passing through 4 meters of air.
Nothing drives light. An object with momentum does not lose its momentum unless it has an interaction in which it transfers momentum to something else. As to why that is the case, no-one knows - it's just the way the universe is. A photon is an object with momentum, so it keeps going forever unless it has an interaction with something else. Joseph H's linked answer covers the interaction with an expanding universe, known as cosmological redshift, which dims$^1$ and cools$^2$ distant light, but does not blot it out or change its direction. To change its direction, light needs to be scattered, which only happens in interactions with matter or gravity, not empty space. To cease to exist, light needs to be absorbed, which only happens in interactions with matter.
1: Light is dimmer which has lower number of photons per second per unit area.
2: Light is cooler which has lower energy per unit photon. In the visible spectrum, red is the lowest-energy color, which is why we call this phenomenon redshift.
A reality-check edit: An equivalent ratio of smaller numbers, like 1 billion light years of intergalactic space to about 30 centimeters of air, would be more in line with real traveled distances through space that really has that density, since if we go too far back in time we have to unwind both gravity concentrating matter in galaxies and cosmological expansion spreading out whatever's left.