How do we know that moonlight is mostly reflected sunlight, and not emitted by the moon itself?

The most obvious reason is that the moon has phases. If it was self luminous then it would glow all the time. Further evidence is that we can use a spectrometer to analyse the light from the moon, which demonstrates that it's reflected sunlight.

Today we know by fact that the surface of the moon is not glowing, instead it shines and that not very good as it is capable to reflect only 12% of the light that hits it. This is happening because of it s surface material which is dark grey, the color of pavement as described by the astronauts. You can see that the moonlight depends on sun s position. That s why we have moon phases when the moon is shining always on the side that faces the sun and we don t see a full moon every day. Also when the sun, Earth and moon are aligned we have "lunar eclipse". In this case, you are able to see earth s shadow covering the moon s surface. That shouldn t be happening if the moon was emmiting it s own light.

Firstly we've been there and looked but also because of the lunar phases and also spectrography

The Lunar phases is a dead give away, Only the side facing the Sun is in "daylight" just like the Earth and all the other planets and planetary satellites, asteroids, etc. Plus the spectrum of light is from the Sun. Rocks do not give off visible light.

If the moon emitted light, it would always appear full.

It's blindingly obvious. You only ever see the parts that are lit up by the Sun. Therefore it can't be emitting its own light, can it?

Dude, the Man in the Moon is actually the Energizer Bunny changing batteries.........

http://phys.org/news/2014-12-mathematicians-umbral-moonshine-conjecture.html
Try this article, I am not sure it is directly valid, or if it is a mathematical metaphor. I hope the link works.

If you analyse Moonlight with a spectroscope you will see that the spectral line pattern is identical to that which we see in the Sun spectrum. That is one clue.

Another is that the Moon has no means of producing energy on its own. The Sun generates its own energy through nuclear fusion in common with all other stars. Presently the Sun is about half way through its core hydrogen burning phase which means it is fusing or binding hydrogen into helium. This process releases a neutrino and (initially) gamma ray photons. By the time they reach the photosphere (visible surface) of the Sun, those gamma ray photons have lost so much energy through collisions that they have moved down to the optical part of the spectrum and we see them as sunlight.

A tiny proportion of those photons are sent out into space in the direction of the Moon where they hit the surface and get reflected towards Earth.

because, when the moon is NOT full, we can;t see that parts that are not light, thus showing us that there is NO LIGHT coming from those parts, which means that the part that is light must be REFLECTING light.