One of the primary reasons for studying the invariant subspace problem is the hope that it will lead to useful structural theories. For example, in their interesting expository paper $\ $ The invariant subspace problem, $\ $ Radjavi and Rosenthal wrote:
It is hoped that knowledge of the invariant subspaces of operators will shed light on their structure. In the case of operators on finite-dimensional spaces, for example, the Jordan canonical form theorem shows that operators
are direct sums of particularly well-behaved operators on certain invariant subspaces. Also, normal operators on infinite-dimensional spaces can be represented as integrals with respect to some invariant subspaces, (their spectral subspaces - this is the spectral theorem). To prove that every operator has a non-trivial invariant subspace might be the beginning of a general structure theory. $\ \:$ On the other hand, $\ $ a concrete counterexample would also be very interesting. To say that $A$ has no non-trivial invariant subspace is equivalent to saying that linear combinations of $\ {A^n\:f\:: n \in \mathbb N}\ $ are dense in the space for each nonzero vector $f$. Representing such an $A$ on various spaces would give a number of approximation theorems.
This paper, combined with Yadav's $\ $ The Invariant Subspace Problem $\ $provide nice expositions on the problem and its rich history. Neither of these introductions are mentioned on said MO page. IMO they provide a much nicer overview of the problem than the little that is said there.