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>The missile is likely to have reached an altitude of about 770km (478 miles)

For reference, the international space station is about 220 miles above earth.

Why does a missile need to do so damn high? Is it because it's just a test? Do all missiles go that high?

My guess is they need to get in orbit in order to go farther and not get shot down as easily.
Getting something high is comparatively easy, getting something fast enough to orbit is a lot harder.

Ballistic missiles follow a ballistic trajectory, there's a high initial impulse and then gravity takes over. Generally, you want the most efficient ballistic trajectory because every kg you save in fuel is 1 more kg for the warhead.