If you take a page of flat notebook and try to stand it up by laying it on its edge on a table it’s nigh impossible. But why does it support itself if you simply add a fold along the middle of the page?
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The other comments are talking about balance but I think you’re really asking why a piece of paper appears to become more rigid when you fold it.
It has to do with surface *curvature*. You can think of curvature as *positive* or *negative* (or zero). The surface of a ball, for example, has positive curvature in both the X and Y directions. If you put your finger on the surface, it curves away from your finger in both directions. A horse saddle or a [Pringle chip](https://www.bakeryandsnacks.com/var/wrbm_gb_food_pharma/storage/images/publications/food-beverage-nutrition/bakeryandsnacks.com/article/2022/05/13/the-world-s-most-expensive-single-pringle/15412813-1-eng-GB/The-world-s-most-expensive-single-Pringle.jpg) has positive curvature in one direction, and negative in the other. That is, if your put your finger in the middle it will curve away from your finger in one direction but towards it in the other. A flat piece of paper has zero curvature – it doesn’t curve in any directions.
The mathematician Guass observed that you can multiply the curvature in one direction by the curvature in the other direction to get the 2D curvature of the surface, and that curvature must be maintained now matter what else you try to do to the surface. If you stretch and twist and shrink and enlarge and push and pull, the curvature must be maintained. For example, think about what happens when you try to make a map of the entire Earth – you **cannot** do it without warping or distorting the map, because the positively curved surface of the Earth *cannot* map 1 to 1 with the zero curve flat surface of a piece of paper.
If you put a crease in a piece of paper, you haven’t changed the curvature of the entire surface, because the negative (or positive) curve you gave it with the crease is multiplied by the still-zero curvature of the paper in the other direction. That curvature *must* remain zero. If you then try to bend the paper again perpendicular to your crease, you would be changing the curvature of the paper, because instead of positive or negative times zero equals zero, you would have a positive or negative times a positive or negative, which would *not* be zero. Therefore, the paper *cannot* bend.
[Relevant Numberphile](https://www.youtube.com/watch?v=gi-TBlh44gY)
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