Where did the waxing crescent get its name?

Answer 1

Waxing is an archaic term used to mean something is growing. So, a waxing crescent means that from the observer's position on earth, the crescent moon is increasing in size. The moon's two main phases are waxing and waning. Waning which means it is getting smaller.

Answer 2

The waxing crescent moon gets its name from the fact that the illuminated portion of the moon appears to be growing or "waxing" larger in size as it progresses from the new moon towards the first quarter phase. The term "crescent" refers to the curved shape of the moon during this phase.