Q:

Why does a light wave fade as it travels through space?

It stands to reason: if it didn't, the entire sky would be covered with stars shining blindingly day and night.
But what causes a light wave (or an electromagnetic wave) to fade if there are no celestial bodies in its way? Or, alternatively, why doesn't it fade sooner? Some stars are visible to the naked eye that are hundreds of light years away. Others, closer, can only be detected with the aid of a telescope. Why? 

A:

There are two phenomena at work here: the inverse-square law and extinction.
Picture a sphere centered on a star. At time $t_0$, a star emits $n$ photons, spread in random directions. At some time $t_1$, the photons are arranged in a sphere of radius $R_1$1. At some time $t_2$, the photons are arranged in a sphere of radius $R_2$, where $R_2>R_1$. This means that the photons are spread out more - and thus, the star appears dimmer at $R_2$ than it does at $R_1$. This is a consequence of the inverse-square law, because the surface area of a sphere is $4\pi r^2$.
Astronomical extinction is the absorption of photons by gas and dust spread throughout space. This can make observations difficult, because gas clouds and nebulae can easily block out most of a star's light.
In really extreme cases, a nebula or a related object will block out all the stars behind it. See, for example, Barnard 68, a molecular cloud:

Image courtesy of Wikipedia user Huntster under the Creative Commons Attribution 4.0 International license.

1 Specifically, at time $t_n$, $R_n=t_nc$, where $c$ is the speed of light.

A:

As the individual photons move away from a point they spread out so the same size telescope or eyeball picks up less of them as you move farther away.
There are more than two phenomena at work here (sorry HDE), in addition to just spreading out based on their initial direction individual photons can be 1. diffracted by moving into more dense space 2. reflected (by any object with a greater diameter than it's wavelength) or 3. absorbed... Especially after picking on him I must encourage everyone to upvote HDE for having a cool picture though!

