Why is belonging not transitive?

A vertex of a triangle belongs to the triangle. A triangle belongs to the set of all triangles. But, a vertex is not itself a triangle.

The difference between $\subset$ and $\in$ is that the former applies to expressions at the same level of nesting and the latter applies to expressions at one level of nesting apart from each other. So when you chain two $\in$'s together you get something at two levels of nesting, which is not in general comparable to a single $\in$. On the other hand, since $\subset$ doesn't change the level of nesting it doesn't have this problem.

This is the idea behind the example given in other answers of $$ \varnothing\in \{\varnothing\}\in \{\{\varnothing\}\},\qquad \varnothing \not\in \{\{\varnothing\}\}. $$

$42 \in \mathrm{Even} \in \mathcal{P}(\mathbb{Z})$ but $42 \not\in \mathcal{P}(\mathbb{Z})$ because 42 is not a set of integers.

$\text{Peter} \in \text{Humans} \in \text{Species}$ but $\text{Peter} \not\in \text{Species}$ because Peter is not a species.