How to erase stains from an image?

If they are dirt on the lens, which is likely the case, they will be stationary, which does help matters.

It is a job for After Effects. You want to make an adjustment layer and draw masks (with feathered edges) that are the shape of those spots and add a Curve adjustment to push those areas darker. Levels may work as well if you find Curves difficult to manage.

What I would do first, however, would be a temporary adjustment layer that maximizes the visibility of the problem by making another adjustment layer with a Levels adjustment wherein I would darken the shadows and lighten the highlights until the problem spots were maximally visible, i.e. had maximum luminosity range from near black to near white within the spots. It doesn't matter if the whites blow out and darks fill in in the overall image, as this would be a temporary adjustment. Then I would make the masked adjustments in another adjustment layer that is between the footage and the "maximize the problem" adjustment layer. This way you can dial in the correction better, and when done, remove or turn off the temporary top layer.