This was too long for a comment.
To elaborate on Hunter's response, there is something you can associate to any curve called its Jacobian variety, whose points can be identified with degree 0 line bundles (these can be identified with formal linear combinations of closed points on the curve whose sum of coefficients is 0, modulo a certain equivalence relation). There is a group operation on the Jacobian variety given by tensoring line bundles (or adding linear combinations), and its dimension is the genus of the curve. Since elliptic curves are genus 1 curves, one might hope that the Jacobian variety is isomorphic to the elliptic curve, and indeed this is the case. So this sort of gives a reason why a group law exists on the elliptic curve.
The catch is that there isn't a canonical isomorphism from an elliptic curve to its Jacobian variety. There is a way to specify an isomorphism once you single out a closed point on the elliptic curve (this is like picking an identity element). So elliptic curves don't really have a group law, it's elliptic curves with a choice of a closed rational point which do.
(I learned this stuff from Chapter IV of Hartshorne's Algebraic Geometry)
