I would like to add a comment about the root reason why bounded cohomology was introduced (see Gromov "Volume and bounded cohomology"). The idea was to create a dual object to homology groups equipped with a natural $\ell_1$-seminorm. This seminorm was used to define simplicial volume - an invariant used in Gromov's proof of the Mostow Rigidity Theorem (see Thurson "Geometry and Topology of 3-manifolds", lecture notes, Chapter 6).

So the boundedness condition allows us to define a seminorm $$\| \cdot \|_\infty$$ on bounded cohomology groups. And there is a duality result:

- If $$\| \alpha \|_1 \neq 0$$ then

$$ \| \alpha \|_1 = \sup \left\{\frac{1}{\|\phi\|_\infty} \mid \phi \in H_b^k(X), \phi(\alpha) = 1 \right\} $$

- $$\|\alpha\|_1 = 0$$ if and only if $$\phi(\alpha) = 0$$ for all $\phi \in H_b^k(X)$.

The other observation is that for spaces with amenable fundamental groups, the bounded cohomology is trivial. The heuristics behind it is that, since cocycles are bounded functions, we can average them using the mean on the group. Even more heuristics is that higher homotopy groups are abelian, and hence they are amenable, so they have no input to bounded cohomology. Hence, bounded cohomology depends only on the fundamental group of our space.

I think that one of the first papers that exploits that heuristics is Ivanov "Foundations of the theory of bounded cohomology", https://link.springer.com/article/10.1007/BF01086634.