At its most basic, a change of variables in a field theory, is analogous to a change of coordinates on a map, e.g. a switch from rectangular coordinates to polar coordinates. Consider what happens to the equation for a circle under such a change. The equation under rectangular (Cartesian) coordinates is x^2 + y^2 = R^2 for some radius R. But if you switch to polar coordinates r, theta, the equation will just be r = R (if the circle is still centered on the origin of the new coordinate system). So a quadratic equation can be replaced by a constant equation, even though it's still the same geometric object.
(Matt Strassler recently blogged about this issue for general relativity, in the context of
the claim that we can equally say the Sun orbits the Earth, because in the geocentric coordinate system, the Sun traces a circle around the Earth. Strassler's riposte is that there is a coordinate-independent reality too, such as the curvature of space-time, and that for one object to actually
orbit another, its orbital acceleration needs to show an inverse-square dependence on distance, something which is invariantly true for Earth going around Sun, but invariantly not true for Sun going around Earth.)
So at the most basic level, a change of variables is simply a redescription. Anything you can do in the new variables, you can do in the old variables too. But new variables can change the form of equations (as for the circle, whose equation went from quadratic to constant), and what Ashtekar's new variables did (
here is the original paper), was to radically simplify the form of some conditions ("constraint equations") that must be obeyed by the gravitational field.
At this point we are still talking classically, about the variables we use to describe different classical states of the gravitational field. When we get to quantum mechanics, we will now be talking about wavefunctions over the whole space of possible classical field configurations. Here I suppose the main significance of the Ashtekar variables is that they make the constraint equations resemble something from Yang-Mills theory, so quantum techniques from Yang-Mills might be applied to general relativity. However, as already foreshadowed at the end of Ashtekar's 1986 paper, ordinary Yang-Mills theory has a background metric (e.g. in flat space-time, the Minkowski metric). Ashtekar's new variables rewrite the metric itself into a Yang-Mills form. So perhaps that is the main challenge here - how do you do a Yang-Mills-like quantum theory, without the use of a background metric?
