Why does the path of the moon in an eclipse curve (relative to the sun) Reviewing several simulations of the solar eclipse, it is apparent that the path of the moon relative to the sun curves. 

Using https://www.timeanddate.com/eclipse/in/usa/kansas-city?iso=20170821 for example, the moon enters at 135 degrees and leaves at 217 degrees direction - that is almost a 90 degree right turn.

Why is the path curved not straight?
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.