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 same reason a table stands with 3 legs and not with 2. When it’s not folded all of it’s edge is along one axis and nothing is stopping it falling along the second axis. Once it’s folded it now has support at 3 points that are NOT in a line with one another, so it’s now supported on both axis.
For a far more interesting experiment, try to use a single sheet of paper to bridge a gap between two tables. A flat sheet will fall through, but add a few folds to make a jigzag and it will be much stronger.
The reason is that a flat sheet cannot take on compound curvature. If you deliberately curve (or fold) it in one direction then it cannot curve also in the other direction. To do so would require it to stretch which it strongly resists.
Paper bends really easily, but it does not stretch easily at all. Bending the paper makes it so that in order to continue bending in the other direction, the paper needs to also stretch (otherwise it would tear). So the paper resists stretching, which has the side effect of resisting bending.
Paper standing on its end is in unstable equilibrium of one of its axes: while in principle it could balance, even the slightest nudge or breath in a direction perpendicular to its face will cause it to accelerate in that direction due to gravity, and thus fall down.
When folded, the paper is now in a stable equilibrium. Any small nudge will cause it to start tipping over, but at that point the center of mass will be positioned so to right the object back to its original configuration. So it takes a substantial push to knock it down.
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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