How does the rotation of the Earth cause seasons?

No, it's the tilt of the Earth that causes the seasons.
When the northern hemisphere is tilted toward the sun, they are having summer and the southern hemisphere is having winter.
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The earth isn't straight up and down with respect to the sun, it rotates at a slight slant towards the sun.  So at times the northern hemisphere is slanted towards the sun which gives it a longer day and the sun is reaching more of its surface, making it warmer.

As the Earth travels around the sun the point that faces changes, and eventually the southern hemisphere enjoys the extra sunlight (ie: summer).

If Earth were exactly straight up and down then we would have no seasons at all.  If one of the poles were slanted directly at the sun then it would have constant daylight/summer and the other half would have constant night-time/winter.

The seasons are caused by the tilt of the Earth's axis of rotation, not its rotation per se.