Moon rise and moon set are not the same as sun rise and sun set. Where I'm at, I can't see the moon. But I'm also sitting at the computer in my windowless basement office so that may have a strong bearing on that fact.
The moon circles the Earth at a different time cycle than the Earth circles the sun, so sometimes it is visible in the sky during the daytime. Sometimes, as well as the "bright" part of the moon, you can see a faint shimmer from the "dark" side of the moon, although it it always the same side facing Earth, just different light patterns. The faint shimmer is light not from the sun, but re-reflected from Earth. It is "Earth shine."
You see it in the day for the same reason you see it at night: it's orbiting the Earth. You just didn't notice it before because the sun makes it hard to see in the day. The stars are still out during the day, too, but the sun wipes them out.