Now what does this mean?

I must say that although I find the thinking in this passage challenging, I don’t find its meaning obscure. Here’s what I think is being said:

“In the same way, the Latin course at St. Gregory’s was taught according to the Direct or Nature Method.”

The author is comparing the way that Natural History was taught at the school with the way that Latin was taught. He says that they were both taught in a way that respected the true nature of the subject being studied and not in a way that sought to control or change that nature. He obviously thinks that the Direct Method or Nature Method respects the nature of language.

“Here, as with the natural world, the object of study was approached with respect for what it truly is, rather than with an eye to its seeming mastery by confining it within the limits of a predatory rationality.”

He continues to draw out the analogy between the Natural History approach to the natural world and Nature Method approach to language. He claims that the standard ways of studying these subjects brings about a “mastery” which is illusory and that this illusion is the result of confining the subject by a use of reason that has the characteristics of a predatory beast. In other words just as the animal stalking its prey sees it only under the aspect of food, so also the limited reason employed by the methods of study being criticised excludes important aspects of their subjects. Judging by words such as “mastery” and “confining”, I would say that the author has Baconian science in mind. Francis Bacon famously wrote of the need to bind and constrain nature through experimentation in order to learn her secrets.

“Thus, since Latin is a language, and language is first a spoken and heard phenomenon that is assimilated as it unfolds within the dramatic story of our lives, Latin was presented to the students in that way.”

Here the author applies to language the principle of seeing things in their unconstrained fullness. He says that language is “first a spoken and heard phenomenon.” It seems that “first” here can be understood in two way: first temporally and first essentially. Spoken and heard language is first temporally because as children we first learn language by listening and speaking before we learn to read and write. It is first essentially because language can exist without writing as it did in Homer’s Greece. When the author says that language “is assimilated as it unfolds within the dramatic story of our lives” he seems to be referring to the fact that language assimilation is greatly aided when interest is aroused by a dramatic situation. This is a principle used in Nature Method courses such as “French in Action” and “Lingua Latina” and it also seems to be at work in the rapid language acquisition of children for whom the world is a dramatic and wonderful place.

“Contrasted to this is the standard, although not traditional, method of Latin instruction in which the language is reduced to the spatial world of a text, and the text presented for decoding with aid of a dictionary.”

Here the author says something about the standard way that Latin is taught now. He says the while this way may be standard, it is not the way that language was always taught. Picking up again the theme of constraint, he says that methods which do not give an adequate place to the spoken and heard nature of language reduce it to “the spatial world of a text.” I think that the author is contrasting the spoken word which happens in time with the written word which only exists on the space of the page. So he is saying that time is the natural environment of language and when we remove language from time by writing it down, we are, in a sense, constraining it. Catherine Pickstock and Walter Ong have made similar points.

I hope this helps to answer your question about the meaning of that paragraph.