The details on how exactly they may have used it to train their model is vague. I believe transfer learning or knowledge distillation are valid techniques based on the inference from other models.

I would also think it'd be an incredibly expensive way to train a model.

Depends. I wonder what is the minimum reasonable amount of different tokens needed to lift up the weights.

You store the output from ChatGPT, you don't run it again every time you do a training step. Generating millions of examples to add to your own training wont cost much at all relatively.