But it rises and sets an hour later each day, which brings it into daylight hours. It shrinks to a crescent, and we lose it in the sun's glare until it comes back. The moon is out as much in day as night, but it's pale instead of brilliant. Look for it, and you'll see it. The moon looks larger on the horizon where we can compare it with familiar things, but the image is the same size high or low.
All sight involves interpretation.
The moon rises and sets every day because the earth turns on its axis. But the moon also orbits the earth once a month. It's that motion that creates phases and brings the moon into the daylight as much as night. The other reason is that when the moon is out during the day, we see a smaller portion of the illuminated side so there is less light being reflected to the earth. It's interesting to track the moon at the same time every day after the full moon.