Could someone explain that term to me, please? All I can find is 'as opposed to city-state'. Does that mean almost all states are territorial states? And what sense does the distinction make in this case?

You're absolutely right: most of all states are "Flächenstaaten" - more or less.
The use of "Flächenstaat" (where it is not obviously meant as the opposite to city-state - "Stadtstaat", e.g. "die deutschen Flächenstaaten Bayern, Nordrhein-Westfalen, Niedersachsen etc." opposed to the "Stadtstaaten Hamburg, Bremen und Berlin") shall put special emphasis on the large area of the state. In developing transportation infrastructure a large territory is kind of obstacle (or challenge as we positive thinkings call it). So even if India has a billion inhabitants and therefore can't just be called empty that billion mostly concentrates in a number of overcrowded places with rather large distances inbetween - so the territory becomes an important characteristic for the issue considered.

Hm, makes sense. Thanks you, HansH!