Why is the stiffness matrix symmetric?

(I'm assuming you're referring to FEM)

The entries of a stiffness matrix are inner products (bilinear forms) of some basis functions. Insofar as you will typically be dealing with symmetric bilinear forms, the stiffness matrix will also be symmetric. In other Words, ai,j = <φi,φj> = <φj,φi> = ai,j.

The issue is closely related to so-called "Gramian matrices" which, in addition to symmetry, have other properties desirable in the context of FEM. I've provided links below.