Since a point can be thought of as a 2x1 matrix, a 2x2 matrix can be thought of as a map from the plane R^2 to itself. Because of the properties of matrix multiplication, this is a linear transformation, and linear transformations have a bag full of cool properties.
There are at least two ways to think about transforming a parabola, to show the effects of a transformation. The first thinks about the parabola being defined by three points. Transform those, fit a new parabola, and voila, transformed parabola. (Blue in the sketch.)
The second shows more what is happening to the whole plane - show the image of every point of the parabola under the transformation. (Green in the sketch.)
Play around with the transformation. Can you make a reflection? Can you stretch the plane or compress it?