Why is there more lunar details visible through a telescope when the mom is near quarter phase than when it is at full phase?

When at the quarter phase, sunlight is hitting the moon 'from the side' from our point of view. Mountains will cast shadows, allowing more details to be distinguished.

Imagine if you are up in a satellite, looking straight down at a large city, like New York. When it is Noon, and the sun is shining straight down on the buildings, all you will see is the tops of the buildings, with not much idea of the heights of the individual buildings. Now in the morning or evenings, the buildings cast shadows, which make it easier to distinguish the heights of the buildings.

When it is a full moon, the middle of the moon is like at 'high noon' and the parts near the edge will be casting shadows away from us (so we cannot see the shadows, anyway).

Frequently, when you see a photograph of the full moon inn a book, it is a composite of two photos, one taken exactly at first quarter, the other taken exactly at last quarter, and combined into one photo of the moon with exquisite clarity.