Learn Russian: Протяжённость этого моста достигает тысячи двухсот тридцати пяти метров. - The extent of this bridge reaches one thousand two hundred thirty-five meters.

Because of the Russian rule for oblique cases. When a compound number is used in any case other than Nominative or Accusative, every individual word within that number must decline. Since the sentence requires the Genitive case, the Nominative тысяча двести тридцать пять transforms word-by-word into тысячи двухсот тридцати пяти.

Why is the number in the Genitive case to begin with? Doesn't the bridge "reach" a direct object?

In English, "reaches" takes a direct object, but the Russian verb достигать (to reach) requires its object to be in the Genitive case. This is true whether you are reaching a physical limit, like a distance or height, or an abstract goal.

How do we get the form двухсот from двести (two hundred)?

Hundreds in Russian are historically formed from two roots, and both halves decline. For двести, the first part declines into the Genitive двух-, and the second part (-сти, meaning hundred) becomes -сот. Put them together and you get двухсот.

Does the noun метров change because the number is in the Genitive case?