I had both of these realizations because I considered whether the opposite could ever happen:
- They have to occur during the night - for the moon to be in the earth's shadow, the earth must be between the moon and sun. Thus, we can only see either the moon or sun. This means it's either daytime, or we see the eclipse.
- They have to occur during a full moon - this one isn't so obvious, but, well, the moon doesn't technically go through light phases and dark phases. It has days (about 28 earth-days long) like we do. When we see a full moon, it means that the side of the moon that we see happens to be the same side that is lit up. This means that the sun and the earth are looking at the moon from the same angle, which is exactly what a lunar eclipse is all about.