I like the topic. Some thoughts:

If you read books like "Pragmatic Thinking And Learning" (a top read for computer programmers, but generally applicable to anything), I think it explains the process very clearly (certainly more clearly than most other sources, except perhaps neuroscience papers on intuition and loss of it in early-stage Parkinson's and Huntington's disease). Visualization, touch, feel, imagining, relaxation, not directing your thinking, stepping away from the problem, etc. are key. The gestalt often comes under the shower, while eating, while walking, and so on, precisely because we are not directing our attention. The book also talks about giving yourself a problem and closing your eyes and then certain visualizations loosely related to the problem may appear. It could be anything... the color or shape of things, voices, activities, really anything. You just have to notice it. It does pop up all of a sudden, when you don't expect it. And I find it most fascinating to engage in this sort of creativity and brainstorming. Due to the non-linear connection-making of certain subcortical structures implicated in this (e.g. basal ganglia), inventions coming from this process are bound to be much more substantial and innovative than sitting on a chair and linearly thinking about something new. After all, many an invention has come out of inventors' dreams during the night, or naps during the day, or otherwise from leisurely doing things that may seem like a waste of time to some.

Now, not understanding these processes our nervous systems are capable of and instead calling it angels and gods and spirits and so on, I think can be very dangerous. Our maps (understandings) of the world direct our actions. If we think there are supernatural beings directing us, or interfering with our affairs, then this will influence our actions. I would think that an incorrect map would lead to incorrect actions. Certainly we've seen lots of both cruelty and ineffective action in the past related to religion/superstition. Curious to hear others' thoughts on this as I may be missing different points of view.