In Einsteins theory of gravity the metric gives a unique torsion free connection called the Levi-Civita connection.
In the Einstein-Cartan theory we allow any connection compatible with the metric and not just the unique one that has no torsion.
Q. Why do we choose a torsion free connection in GR?
Q. What does torsion represent physically in the Einstein-Cartan theory?
You asked a two part question, and I'll give you a two part answer.
First: outside of matter, the connection is torsion-free. This is the case in General Relativity, as given either on a Riemannian geometry with the Einstein-Hilbert action or as transcribed from there directly onto a Riemann-Cartan geometry into the Palatini action. It is also the case for Einstein-Cartan gravity, as given by the Einstein-Cartan action on a Riemann-Cartan geometry. For both cases "torsion does not propagate" outside of matter. (To clarify a bit: "fields" are included, here, under the header of "matter", so in a sense you're always "inside" matter of some kind).
General Relativity was formulated without torsion, entirely as a matter of historical precedent (that is: bad timing), as the very concepts of Riemann-Cartan geometry, torsion and spin, did not exist in any well-known form at the time the Einstein-Hilbert action was laid out.
Second: torsion can have an effect on the law of inertia/gravity, itself, and on the very definition of what constitutes "inertial" or "free-falling". Unlike Riemannian geometry, there is now a distinction between the geodesic curves and auto-parallel curves. They're both defined by equations that look like the geodesic equation, except the one for the geodesic trajectories works with the Levi-Civita connection, while the one for auto-parallel trajectories works with the connection that's native to the Riemann-Cartan geometry. The difference between the two involves torsion.
This naturally raises the question: does the law of inertia/gravity mean matter follow the longest-time trajectory (the geodesics) or the straightest trajectory (the auto-parallels)? If it follows the straightest trajectory, then there can be deviation from geodesics, in the presence of torsion.
So, which one is it? Geodesic or Auto-Parallel? Here: Test Bodies: Auto-Parallel or Geodesic?
