Lots of ways to do it.

One way you might could try is by using an extruded shape along a path (Surfaces > Extrude. Ask if you don't know how to use it), but check the "Partial" option in the option box.

You'll get 2 extra attributes in the Extrude options in the channel box. One controls the length of the surface along the profile curve, the other the path curve. You can animate the path curve option from 0 to 1 to simulate the shape "growing" along the path.