Why does the path of the moon in an eclipse curve (relative to the sun)
You have misunderstood the website. When the eclipse begins, in the morning, the sun will be in south-east, at a bearing of 135 degrees. 

At maximum eclipse (at about midday) the sun will be almost due south, at a bearing of 178 degrees.

When the eclipse ends, in the afternoon, the sun will be in the south-west, at about 217 degrees bearing.

The sun and the moon will move in the sky, due to the rotation of the Earth, this says nothing about the straightness of the path of the moon relative to the sun.