Why can you see the moon during the day?

The moon is above the horizon during daylight for a large portion of its monthly cycle, and it reflects enough sunlight to remain visible against a bright blue sky. Nothing unusual is happening. The moon rises and sets roughly 50 minutes later each day as it orbits Earth, so its position relative to the sun shifts continuously: a full moon rises around sunset and sets around sunrise, keeping it a nighttime object, but a first quarter moon rises around midday and is high in the sky all afternoon, and a last quarter moon is prominent in the morning sky. Those daytime appearances are ordinary, and the moon is visible in daylight on a great many days of the year.

What makes it possible to see is contrast. The daytime sky is bright because the atmosphere scatters sunlight, but the moon's sunlit surface is brighter still, so it shows through, appearing pale and washed out rather than the vivid white of night. This is also why you rarely see a crescent moon in daylight, it is both dimmer and positioned close to the sun in the sky, which drowns it out, and why a nearly full moon in the afternoon is easier to spot. There is no scientific mystery here, just a widespread assumption that the moon belongs to night, which is a cultural habit rather than an astronomical fact.