Signed area refers to the area between a function and the x-axis, with the sign determined by whether the curve is above or below the x-axis. When the function is above the x-axis it's area is added where as when it is below the x-axis it's area is subtracted.

Example

The area shown in green as added to the signed area, where as the area shown in red is subtracted from the signed area.

Because is an odd function, the area below the x-axis on the left is exactly the same as the area above the x-axis on the right hence: