However, simple frame-by-frame segmentation will probably not be enough to get temporal consistency, so for each frame's segmentation they probably take previous and following frames into account.
For a deep learning approach, I would start by looking into literature on semantic segmentation. Here is a blog post I just found which gives an intro: [1]
With state-of-the-art models (e.g. DeepLabV3) and a good dataset of foreground/background segmentations, the results could be of useful quality already.
The next step would be to look into literature on image matting (e.g. deep image matting [2]) which instead of trying to classify each pixel as foreground/background, regresses the foreground colour and transparency.
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[1] https://divamgupta.com/image-segmentation/2019/06/06/deep-le...
I have some knowledge of creating an OCR program using deep learning from the last online course I took, but this looks like a very different beast and so it would be great fun to learn