Earth rotates fast in the sense that the linear speed of the rotation at the equator is fast. It does not rotate fast in the sense that it makes a full spin quickly (it doesn't; it makes a full spin in - from a distant perspective - 23 hours and 56 minutes).

If you were floating just above the equator and not co-rotating with the Earth, you would see the land below you flying by at an extremely fast pace (about 1.5x the speed of sound). But if you're in space, you're quite far from the surface of the Earth - at least a hundred miles or so - so that speed doesn't look too too fast.

In the very best case, where you're 100 miles above the Earth, not in orbit, and not co-rotating at all, you'd see the Earth move at a speed of about 0.2 degrees across your field of view per second. That isn't nothing, and you probably would notice it if you were paying attention, but it's easily lost in other motion. For comparison's sake, a finger at arm's length covers about 1 degree, so an object directly below you would take about 5 seconds to cross the width of your finger at arm's length. Or, put another way, it would move across your field of view at about the same rate as a person walking slowly on the other side of a football field from you.

Fundamentally, it looks slow because of parallax: faraway objects don't move across your field of view very quickly even when they're moving very fast.

In practice, though, you never get even that much, because:

• If you're in orbit anywhere near the Earth, you're actually moving faster than the Earth rotates (by a lot!), so you mostly see your movement over its surface. You're also higher than the 100 miles in our example: both active space stations (the ISS and China's Tiangong) orbit at about 250 miles up.
• If you're not in orbit, you probably just launched from the surface of the Earth, and you still have the horizontal momentum you started with, so you're still more-or-less co-rotating with the Earth.