A fushime means an “important point of transition in life”
To understand the concept, picture a piece of bamboo.
A bamboo stalk is divided into fushime (nodes) and growth from node to node results in the fully grown plant.
So it is in life, where each specific period builds on the previous one.
The idea of continuity is thus an important part of the concept of fushime and it explains why, when the Japanese complete something, they like to hold ceremonies and thank people, before moving on to the next stage.