Question

why does the sun appear to move across the sky

Answer 1

Well it appears to move across the sky since the earth rotates around the sun.

Answer 2

The Earth is spinning. About once every day. You are on the Earth so you are spinning too. You have to keep turning your eyes to look in the direction of the Sun. You can't feel the Earth spinning so your brain decides that you have to keep moving your eyes because the sun is moving.