The apparent curvature of the moon's transit across the sun is primarily due to the change in perspective of the viewer as s/he is rotated to the east by the earth's normal daily-cycle rotation - approximately 15 degrees absolute per hour.
At the beginning of the eclipse, the viewer is standing "upright," and sees the moon touch the sun at approximately 1 O'clock. Over the next three hours, as the eclipse progresses, the viewer is "laid on her/his side" relative to his/her initial orientation due to the rotation of the earth; as if tilting his/her neck to the east. Since most of us were looking (essentially) towards the south for this eclipse, it was as if we had tilted our head to the left, so the exit of the moon - instead of appearing at approximately 7 O'clock as expected for a linear transit, the exit point appeared to be at 9 O'clock.
The magnitude of this effect is dependent upon several variables, including the viewer's latitude during observation.
The effect is also exaggerated by the fact that the sun transited through its peak azimuth between the beginning (rising towards its highest azimuth - approximately 63 degrees) and the end of the eclipse (setting away from its highest azimuth). While the earth itself only rotated approximately 45 degrees absolute during the 3 hour event, the apparent rotation of the moon's transit line across the sun was approximately 60 degrees (in the Kansas City simulation).