Affine space means a vector space in which the origin is not preferred. Its automorphisms are the group of affine linear transformations over $k$ (the usual linear transformations with respect to some (arbitrary) choice of origin, together with translations).
In algebraic geometry, one often encodes geometric objects in slightly complicated ways, and the preferred method of encoding is time-dependent, e.g. in that what Weil meant by a variety, and what it typically meant now, are not literally the same mathematical object, although the are both supposed to refer to the same Platonic reality.
Thus, one shouldn't think about $\mathbb A^n_k$ too literally in terms of its points when comparing with the Wikipedia discussion of affine space (since what its points are will depend on what foundations you use!). It does encode the geometric concept of affine space (as in the linked Wikipedia article) in the context of algebraic geometry. For example, its automorphisms are precisely the group of affine linear transformations over $k$. [Correction: As Mariano points out in a comment below, there are many non-linear automorphisms of $\mathbb A^n$; what I should have written is that the automorphisms of $\mathbb A^n$ which extend to $\mathbb P^n$ --- so these are the automorphisms of $\mathbb A^n$ that extend to the hyperplane at infinity in some reasonable sense --- are precisely the affine linear automorphisms.]
If you do insist on making the connection precise on the level of points, then
one way to say it is that if $\Omega$ is any extension field of $k$, then
the $\Omega$-valued points of $\mathbb A^n_k$ are naturally an $n$-dimensional affine space over $\Omega$.