How did Rammus get this fast?

The last soft cap for movement speed is at 490 u/s as @VanBuzzKill mentioned in his comment.

This means while the movement speed gained from spell is heavily reduced, it is still possible to achieve insanely high speed numbers.

In your video Rammus got a Righteous Glory, a Dead Man's plate, a Yomuu's, Boots of Swiftness and a Runic Echoes enchantment. In addition to that he got help from Zilean (99% MS) and Homeguard (150% MS).

The most important thing about reaching this much MS without much effort is Rammus's Q which grants him an additional 275% MS increase that works multiplicatively (most percentage increases for movement speed work additively, this one doesn't). This is the main reason why it's so easy to reach high speeds with Rammus in a normal game.