I haven't read anything in this thread in order to keep my answer clear.
This is a basic reinterpretation of the macroeconomic identities for a two-sector economics which is all about money flows and the fact that for each part of the economy (households, firms), the sources are equal to expenditures. The money is always circulating. Do not pollute your mind by any real world institutional factors, this is a model, it abstracts from all those tax shields and whatever unless it specifically says so.
The thing that got you confused is that the book is apparently very basic. If it says exactly "Households (via their savings) *supply* capital to firms who *demand* it.", it apparently abstract from the financial sector as well, which is an unnecessary abstraction. The book says that firms do not save money themselves, therefore their outflows are equal to wages W and residual profit π, both going to households (owners are households as well); the inflows are investment I and consumption C. The balance of households is thus C+I=W+π and the balance of firms is W+π=C+I. All is nice and balanced but very crude.
Let's add the financial sector. The financial sector in an intermediary, which enables efficient allocation of savings to investment. Abstracting from it is therefore bad and you should sue your book's authors for fooling you.
Firms still have inflows equal to C+I and outflows to households equal to W+π but this time, π is much lower since the firms also have outflows to the financial sector. These outflows are Sf (savings of firms) and Df (depreciation funds). The households still have an outflow C towards firms but instead of I to firms, they send Sh (savings of households) to the financial sector. The inflows of the financial sector is therefore Sh+Sf+Df and the outflow is I, investment funds flowing to the firms.
Therefore, your book basically presents a limit case of the latter model. Actually, scrap that, it doesn't present even that. It assumes that households themselves are perfectly capable of allocating the investment to the firms and therefore the finacial sector is redundant. Now, about the limit case I thought for a while it represented: if in the latter model the savings of firms and depreciation funds were equal to 0, the whole model would work and your book would be right. However, that is not the most probable scenario. In a similar manner, we could say that the households do not save anything and that their whole outflow is C. The model would still work but it's explanatory power in regard to real world would be even weaker than the two-sector model already is.
In conclusion, whether your book proposes that there is no financial sector or that the firms only have outflows W+π, they are unnecessarily weakening the explanatory power their model could have. The additional labour the students would need to use to learn the broadened model is minimal. Plus, it confused you by its simplicity so I could probably say that simpler doesn't always mean simpler :)