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.