Let's start with a simpler example. Take the standard xy-coordinate system and draw a line segment from the origin to an arbitrary point (x, y). The length of this segment will be sqrt(x^2 + y^2). Now, suppose that we want the coordinates of the point on this segment that is a distance of 1 from the origin. The coordinates of this second point would have to be (x / sqrt(x^2 + y^2), y / sqrt(x^2 + y^2)). One can verify that this second point does have a distance of 1 from the origin.
The same principle applies for (3), the difference being that we are operating in n-dimensional space instead of 2-dimensional space. In this case, view sqrt(Σ (x_i - xbar)^2) as the distance the point (x_1, x_2, ..., x_n) is from the point (xbar, xbar, ..., xbar), or, if you prefer, the distance the point (x_1 - xbar, x_2 - xbar, ..., x_n - xbar) is from the origin.