Learn Russian: Протяжённость дороги достигает километров трёхсот восьмидесяти. - The extent of the road reaches around three hundred eighty kilometers.

The verb достигает (reaches) requires its object to be in the genitive case. In Russian, when declining multi-word compound numbers, every word of the number must decline. So the nominative триста восемьдесят changes completely: триста becomes трёхсот, and восемьдесят becomes восьмидесяти.

How exactly do the internal parts of these numbers decline?

For compound numbers ending in tens (50-80) and hundreds (200-900), both halves of the single word decline. In 300, три becomes трёх and ста becomes сот. In 80, восемь becomes восьми and десят becomes десяти. Combined, you get трёхсот восьмидесяти.

Why is the noun километров placed before the number instead of after it?

Placing the counted noun before the number is a common Russian trick to express approximation. Because километров comes first, it implies around 380 kilometers without needing an extra word like примерно (approximately).

Does swapping the word order for approximation change the grammatical case of the noun?

No, the grammatical cases remain exactly the same. The verb достигает still forces the entire phrase into the genitive case. Whether you say километров трёхсот восьмидесяти for an approximate amount, or трёхсот восьмидесяти километров for an exact amount, the noun and number endings do not change.