The simplest explanation is given by Newton's theory of Universal Gravity.
He showed mathematically that if gravity is an instantaneous inverse-square force acting in proportion to mass, and assuming his three laws of motion, then two spherical bodies will orbit each other in ellipses around the common centre of mass.
Keplers' laws only apply in those conditions, and in reality, the solar system has more than two bodies. And Newton's laws are only an approximation to General Relativity (and quantum mechanics). Fortunately, in many situations, such as a planet orbiting a star, Kepler's laws are still a very good approximation, because Newton's laws are a very good approximation to relativity (at everyday speeds), and because the interactions between the planets are much smaller than the interactions between the sun and each planet. So it is a very good approximation for the motion of a planet to apply Kepler's Laws
Alternately, the motion of the planets in the sky fits the motion predicted by Kepler's laws. This provides empirical confirmation of Kepler's laws, and so indirectly it supports Newton's theory of Gravity.
In extrasolar planetary systems, Newton's laws will still hold, as an approximation to Relativity. And as systems in which there are significant interactions between three bodies at once are highly chaotic, and unstable, such systems won't exist for long.
Therefore one would expect that in all systems Kepler's laws will hold (in an approximate way as they do in the solar system).