two uncommon verbs describe the moons act of getting larger and smaller.
Waxing - getting larger
Waning - getting smaller
So 'the moon, waxing towards full' means 'the moon, which was getting larger as the month passed, and was now almost a 'full moon'
The meaning of Pulsed is separate. Consider the meaning if we remove the subclause:
Starlight showered and the moon pulsed against the night sky.
Pulsed, in the context of light, means quickly alternated between bright and dim. I don't quite understand the authors meaning, but that is the literal meaning of the sentence.