This slide is intended to show the connection between span and linear map. It highlights the fact that the plane formed by all the vectors resulting from linear map transformation, f, is the same as the span of a set of vectors. (in this case, it is a1 and a2)
Similarly, image of a function could also be visualized as the span of some collection of vectors.
The example shown seems to suggest that, in general, if vectors a1, a2, ..., aN form a basis for the domain of a linear map f, then f(a1), f(a2), ..., f(aN) should form a basis for the image of f.