Neglecting the relatively tiny gravitational force, the only forces on the piece of string are the tensions at both ends, acting parallel to the string end. They have magnitude equal to the tension in the string, but are out of balance unless the piece of string is straight. The green arrow shows the sum F of these two forces, this is the force accelerating the piece of string.
If you use the slider to put the center of the red piece of string exactly at the node (quiet spot), that piece of string feels no net force. So why does it move at all?