Why do folded concentric circles and rectangles form a hyperbolic paraboloid? Here is a "self-forming" origami that I made from folding concentric circles - it would also happen if I folded concentric rectangles. How can the fold shapes such a saddle-like geometry?

 A: You might want to look at (Non)Existence of Pleated Folds: How Paper Folds Between Creases by Demaine, Demaine, Hart, Price and Tachi from 2011. It analyses several aspects of the objects you are asking about. In particular (and for lack of time just judging from the first few pages), it claims that the version made from concentric rectangles cannot actually exist in a strict mathematical way, at least not with only the intentional creases. Additional slight creases are needed to make this work. A round version apparently would fold without creases, so your photo looks like it might agree with mathematical models of paper folding.
The authors of that paper are also good pointers for further information. Erik Demaine in particular hosts a page on Hyperbolic Paraboloids and also wrote a paper on Curved Crease Origami. I haven't researched the others as thoroughly.
A: Here is a very quick explanation. I'm sure the situation is analyzed in much more depth in the references in MvG's answer.
By pleating you have reduced the distance $r$ from the center of the disk to the circumference, but the length of the circumference $c$ has remained the same. There isn't room for the length $c$ of paper along the boundary to fit in the $2\pi r < c$ amount of space a flat disk would have, so it has to buckle.
In fact your shape is analogous to a disk in hyperbolic geometry, whose circumference is greater than $2\pi r$, and so it curls up when embedded in Euclidean space. The same thing happens in nature in leaves, jellyfish tentacles, and dried fruit.
If you had created pleats along radial lines instead, you'd have $c<2\pi r$ and the paper would form a cone, which is a rough approximation of spherical geometry.
