This is an addendum to the information that you already have. Firstly, there are three sources of substantial results on Köthe spaces—unsurprisingly the first volume of his monograph (which also has useful references), Grothendieck‘s dissertation (and his textbook on topological vector spaces) and the monograph by Valdivia suggested above. Secondly, as already mentioned, they are a useful source of models for spaces of test functions and distributions. There is a useful unified approach to this which is, I hope, worth mentioning. Suppose that we have an ubounded, self-adjoint operator $T$ on Hilbert space. Then the intersection of the domains of definition of the its powers has a natural Fréchet space structure and many of the classical spaces of test functions can be obtained in this way. A dual construction leads to the corresponding spaces of distributions. The connection with Köthe spaces is provided by the fact that if the spectrum of $T$ is discrete ,i.e., consists of a sequence of eigenvalues, then this space (and its dual) are Köthe spaces which can be described explicitly in terms of this sequence. An advantage of this approach is that it can easily be generalized to give, e.g., ultradistributions, Roumieu spaces and Sobolev spaces, even of infinite order.
One example of the benefits is that this approach can be used to give transparent proofs of important results—the classical case is one of the celebrated kernel theorem of Laurent Schwartz.