There are a number of ways of accomplishing what you're seeing, depending on how you'd like it to behave. I'll explain some of the simpler methods of modifying a Bezier curve via point on curve manipulation.
The first thing to do is figure out the parameter value (t) where the user clicked on the curve. This is generally going to be an approximation. If you're doing pixel or sub-pixel rendering of the Bezier, then just record for every pixel what the t value was and use that. If you're tessellating into line segments, see which line segment is closest, find the t values of the two end points, and lerp the t value according to the distance along the line.
Once you have the t value, you can plug it into the Bezier curve equation. You'll end up with something of the form:
P = k0 * P0 + kb * P1 + kc * P2 + kd * P3
where k0, k1, k2, and k3 are constants for a given t. I'll call them contributions, or more specifically the contributions of control points to the point on the curve P(t). A nice property to remember is that k0+k1+k2+k3 = 1.
So, let's say you have a vector V = P' - P, where P' is the new position and P is the original position. We need to move some of the control points to get P' where it needs to go, but we have some flexibility about which of the control points we want to move. Any point with non-zero contribution can be used, or some combination.
Let's say the user clicks on the curve at t=0. In this case, only k0 is non-zero, so
P0 := P0 + V
will produce the correct result. This can also be written as
P0 := P0 + k0 * V
In the general case where all of the contributions are nonzero, you can apply the same transformation to each of the points, which will have the effect of a very smooth, spread-out deformation.
Another option is to simply move the control point with the maximum contribution the entire distance. I think the equation to use would be something like
Pmax := Pmax + 1/kmax * V
but either way it boils down to looking at the contributions at a given t value, and moving the control points so the new point lies in the desired location.
This approach is fairly general, and works for NURBS and most other splines, even surfaces. There is another method that's fairly common that uses Greville Abscissae, which pins as many points as possible, but in my experience it's too easy to get oscillation.