What is meaning of Advance of ships?

This is the distance the ship travelled in the direction of the original heading, measured from the point at the moment when helm was  first applied to the turn.

In other words we can explain advance as:

Advance is define by the forward motion of the ship, from the moment that the vessel commences the turn. It is the distance travelled by the vessel in the direction of the original course from commencing the turn to completing the turn. It is calibrated between the course heading when commencing the turn, to when the vessels head has passed through 90°.