Assuming that the grid & coloration on the surface need to be maintained, one approach would be to create the fully-revolved model, then by a process of duplication and mesh editing, create however many intermediate shapes are needed to reduce the volume to its beginning form. These can then be animated to “grow” the full volume by a number of means. Simple object replacement is one option, and would give the best impression of the volume sweeping through its revolution. Another would be to do passes on the individual objects and blend them together sequentially in the VSE, where more stylization could be introduced. If the camera does not move, single images could be made of each step in the revolution and then the animation done entirely in the VSE with hard cuts between the images.
I don’t think shape keys, the usual method for progressive shape changes, would work here because the initial forms would have to have as many vertices as the end form, leading to a lot of crowded geometry in the first steps of the sequence.
If you can get by with individual segments rather than a fully continuous volume as it grows, an array modifier plus curve modifier might serve but would be tricky to set up for perfect spacing of the segments along the curve of revolution.