Take a look at quiz numero 7!! In addition to using barycentric coordinates to sample from a given texture, we can also use it to determine whether points are inside or outside of a triangle. So during the triangle rasterization phase, we are already essentially computing barycentric coordinates that can be re-used later on during the texture mapping stage.
How exactly do we "get them for free"?
Take a look at quiz numero 7!! In addition to using barycentric coordinates to sample from a given texture, we can also use it to determine whether points are inside or outside of a triangle. So during the triangle rasterization phase, we are already essentially computing barycentric coordinates that can be re-used later on during the texture mapping stage.