Earth loses water vapour from the atmosphere and it blows away into space on the solar wind. Some of it condenses onto the coldest parts of the surface of the Moon.
Comets contain quite a lot of water and bombarded the Moon for a long time during the early parts of the formation of the Solar System.
The source of water on the Moon is debated. Some calculations indicate the lack of an atmosphere on the Moon would result in the loss of virtually all the water sourced from the comets in the Late Heavy Bombardment.
The recent probe targetted the bottom of a deep crater near the South Pole of the Moon because this place is very cold. The hope was that some water still remained due to the cold and despite the lack of atmospheric pressure.
A significant amount of water is still deposited on the Earth every day from space. Some would also fall onto the Moon but the the collision energy results in most of it being immediately lost.
One thing for certain is that it didn't come from the "ash" in the collision that formed the Moon. All the water in the Earth/Moon system would have been lost to space in the collision. So much energy would have been the generated in this event that all the rock melted. Any water would have boiled off into space.
All the water on both Earth and Moon came after this event which happened before the Late Heavy Bombardment.
As the water was found on the dark side of the moon and had been there for millions of years it looks like it was delivered via comets as mentioned above. It may even suggest that life on Earth came from comets or asteroids. Maybe those comets are still whizzing round the solar system with potential life on them. That leads to another question: where did comets originate from?