Essentially, winter squash and also sweet potatoes contain excess water when first harvested, even when they are mature. Curing ( storage in a warm dry place for a couple of weeks) allows the dispersal of the excess water and concentrates the natural sugars.

Thanks farmerdilla. I'm still wondering if there are color clues or do I just wait for the vine to dry up or is that overdoing it, when picking pie pumpkins. It's pretty clear to me that in the past I've been picking squash and pumpkins before they have cured or worse. I dug my first sweet potatoes last week which had been in the ground maybe as long as two years (didn't do anything first year). One was as big as a bowling ball. Didn't let them sit to dry, just roasted them and we really enjoyed them. But I would say I've had much more sugary ones (ideally).

While I usually wait for the vines to show definite signs of passing, Hard rind and dying stem are good indicators. Many of the "pie" pumpkins are small orange types. when they turn orange they are ready to be picked. Flavor does improve with curing. Commercially most pumpkin pies and mixes are made with C. moschata types ( members of the butternut group which are tan or terracotta when ripe) Somes of these are very slow to color so color is not always a good indicator of when to pick. The other group used in the industry are C. mixta (Cushaw type) which show little to no color change when ripe.
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Thanks again farmerdilla - so I'm guessing this unfamiliar and charming candy-striped squash pictured is an example of C. mixta, cushaw type? I don't know about commercial offerings but was raised eating scratch-built pies made from "cheese pumpkins" which are cinderella-shaped but butternut-colored. Since leaving the garden state I've had progressively more trouble, even through farmers markets coast to coast, obtaining pumpkins or squash tasty enough to satisfy those pie cravings. A couple years ago I grew Marina di Chioggia that were excellent for pie and that's how they were consumed, for some reason the plants all disappeared this year. Maybe the handsome warty Thai squash will find it's way into a piecrust this year!

Yes, that is the Gold Striped Cushaw, not as popular as the Green Striped but much prtettier. The cheese pumpkins so named because the shape reminded the old folks of a wheel of cheese are very popular for pies. Long Island and Tan cheese are the more popular, but there are quite a few of them in C. moschata group. The Muangs are an Asian version. Here is another the Beung Carn.
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Thanks for these lovely squash photos farmerdilla. I'm sorry my turban squash have once again failed to germinate so my fall garden decor will lack them, but the asian warty squash are curing on the vine in the garden slowly changing color as you illustrated, and I look forward to gobbling them up.