Learn Hardcore Russian: В озере глубже, чем в реке. - It is deeper in the lake than in the river.

Russian often omits the present-tense form of to be (есть) in ordinary statements. So the meaning is understood as (It is) deeper in the lake than in the river without any verb being written.

What exactly does глубже mean here, and what form is it?

Глубже is the comparative form of глубокий (deep) / глубоко (deeply). In this sentence it functions like a predicate meaning deeper (i.e., the depth is greater). Russian commonly uses the comparative like this without an explicit noun: (там) глубже = it’s deeper (there).

Why does the sentence start with В озере and not Озеро?

Because the sentence compares depth in places, not the two bodies of water as objects.

В озере глубже, чем в реке. = It’s deeper in the lake than in the river (talking about the water depth there).
If you say:
Озеро глубже, чем река. = The lake is deeper than the river (treating them more like objects being compared overall).

What case is озере / реке, and why?