Fringing of electric field

How is the field produced? By charges on the surface. If you go to the quantum frame, it is excess electrons on one plate and excess positive charge (holes) on the other plate. Think of the electric field generated by an electron. It goes radially out. In an infinite plate capacitor the addition of the fields, because of symmetry becomes vertical. Given dimensions, the electrons at the edge will be having lines radially out, the positive charges on the other plate will go to meet them again radially out, because that is the geometry of the point charges. On the side that is in the air, there are no fields to add towards the vertical and the shape is like the field shape in two dimensions of a pair of +- , in the line perpendicular to the edge.

There are many ways of answering your question but I think one of the simplest is as follows:

Assume there is no fringe field when a capacitor is storing charge.
Move a positive charge from the outside of the negative plate to the outside of the positive plate.
Since there is no fringe field, the work done in moving that positive charge between the plates is zero, but that cannot be so as that would imply that there was no potential difference across the plates.

With a fringe field present and weaker than the field deep inside the capacitor, move a positive charge along a fringe field line from the negative plate to the positive plate.
The potential difference between the plates is $-displaystyle int^{large +}_{large -} vec E cdot dvec s$.
Although the fringe field is weaker than the field deep inside the capacitor, the path length is correspondingly larger which results in the same potential difference.
With the field curving inwards you would get a larger field strength and a larger path length, ie. a larger potential difference.