In The History of the Peloponnesian War by Thucydides, what light do the speeches like the one titled "The Athenian Case" shed on Athenian political practices and policies, Athenian democracy, and the effects of war on the Athenian population?

I'm not quite sure which passage you are referring to when you ask about the "Athenian Case." In my copy of Thucydides, I've never exactly seen that term used specifically (there are at least two candidates that immediately spring to mind), and even a Google search with the words "Athenian Case Thucydides" returns very little to clarify this question. Regardless, Thucydides, when writing about the nature of war and political policy, has a very pragmatic analysis, by which moral and ethical considerations are ultimately powerless when juxtaposed against military and political might. This analysis is invoked by the Athenians, who are ultimately motivated by their own political self-interest, and a very pragmatic approach which could be labeled a form of realpolitik. For all that the Athenians are often celebrated as a bastion of democracy and Greek culture, when it came to their political calculus, they were self-serving, brutal, and deeply amoral.
There are at least two encounters in History of the Peloponnesian War that advance this thesis. The first, found between 1.66 and 1.87, details a conference held in Sparta on the subject of Athenian aggression. Athens sent their own representatives to defend the city-state against the charges being laid against it. In this speech they invoke the memory of Marathon and the Persian War, and the Athenian role in throwing back the invasion. After this, they defend their acquisition of empire on three grounds: "security, honour, and self-interest." From here, they note, that "it has always been a rule that the weak should be subject to the strong" and that Sparta is just as much subject to this rule as Athens itself and is little different in this respect. For Athens, it is natural that the strong dominate those weaker than themselves and that Athens deserves praise for being more magnanimous a master over its subjects than it needs to be. Finally, the Athenians warn Sparta that, should it destroy the Athenian Empire and take its status as dominant Greek hegemon, it would soon see itself become a target of hatred as well.
The second encounter I'm thinking of is the Melian Dialogue (one of the most famous scenes in Thucydides), a conversation between the Athenians and the Melians (whose colony the Athenians had under siege), on the subject of surrender. Here we see many of the same themes and ideas expressed earlier by the Athenians at Sparta. Famously, the Athenians repeat their same underlying political thesis: "the standard of justice depends on the equality of power to compel and that in fact the strong do what they have the power to do and the weak accept what they have to accept." This is an encounter between the stronger (Athens) and the weaker (Melos) and opposes a political calculus focused on a kind of realpolitik against one grounded in moral virtue. The Melians appeal to notions such as justice, friendship, perhaps even piety, while the Athenians continuously reject their appeals, holding fast to their threats of Melian destruction. The Melians refuse to surrender and the war continues. In the end, Thucydides tells us, the Melians were conquered by force, with the adult males being killed while the rest of the population was sold into slavery.
Citation note: this answer was written with reference to: Thucydides, History of the Peloponnesian War (Penguin Classics). Translated by Rex Warner. New York: Penguin Books, 1972. Athens' speech to the Spartans could be found in the chapter labeled "The Debate at Sparta and Declaration of War" which runs from pp. 72-87. The Athenian speech itself is found on pages 78-82 (1.73-1.78). The Melian Dialogue, on the other hand, was found between pages 400 and 408 (5.84-5.116).