The tangent of a Bézier curve is the line which connects the path point to the control point. The curve follows along this line as it leaves the path point, thus it is actually the tangent of the curve at the path point.
With this function, these tangents are made visible, giving a more precise overview of the construction of the path. Activating the icon shows the tangents, otherwise they are invisible.
Hint: Use this switch in Edit Grid mode.