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Any traveler's observation of a beauteous waterfall requires her to organise her observation by time and space; by time because she must think of time as increasing to observe (the beauty of) the falling water; by space because she knows to observe the waterfall away at a safe distance.

This is a great example of exactly what Kant means by the term 'Transcendental philosophy'. According to Kant (see the chapter titled 'Transcendental Aesthetic' in *The Critique of Pure Reason'), the true aim of philosophy is to understand a thing, not by analyzing that thing itself (for this is the role of science) but by thinking about what it is that the existence or possiblity of that thing presupposes. From the introduction to the Critique of Pure Reason:

I entitle transcendental all knowledge which is occupied not so much with objects as with the mode of our knowledge of objects in so far as this mode of knowledge is to be possible a priori [Kant's emphases]

In the Transcendental Aesthetic, Kant shows that space is the form of appearances and time is the form of (inner) experience. They are both required in order for any experience to be possible. So you are right in noting that

Even when viewing rhapsodies or kaleidoscopes, do not humans interpret them spatially and temporally?

The precise relationship between perceptions and space and time (or perhaps more accurately, experience and space and time) is in fact a logical one:

Experience is possible => Space and Time exist

where the '=>' means logical implication, in a 'special' form which we might call 'transcendental deduction'. So, if we can grant that we can have experience, then we can conclude that Space and Time exist and pre-exist things or actual bodies, which was quite a novel concept at the time, as space (following Leibniz) was considered to be nothing other than a relationship between bodies.

In regards to the second question about the distinction between 'mental pictures' and 'organizing principle or rule': the author seems to be getting at Kant's distinction between intuition and concepts. Intuitions are original and relate directly to appearances, being caused by sensation of them. Concepts, on the other hand, are derivative and are derived from intuitions.