Definition of an affine subspace

If you take a subspace and shift it away from the origin, you get an affine subspace.

In other words, an affine subspace is a set $a+U=\{a+u \;|\; u \in U \}$ for some subspace $U$.

Notice if you take two elements in $a+U$ say $a+u$ and $a+v$, then their difference lies in $U$: $(a+u)-(a+v)=u-v \in U$. [Your author's definition is almost equivalent to the one I've given above. The author mistakenly says "for all $x,y$ when it should be for any fixed $x$ all $y$ lie in there iff $x-y$ lie in the subspace.]

If you are familiar with a bit of modern algebra, affine subspaces are just elements of quotient vector spaces. So for example, given $U$ a subspace of $V$, the set $V/U = \{ a+U \;|\; a \in V\}$ is the quotient of $V$ by $U$. It is a vector space itself (briefly, its operations are $(a+U)+(b+U)=(a+b)+U$ and $s(a+U)=(sa)+U$).

More concretely, the affine subspaces associated with $U=\{0\}$ are $a+U=\{a+0\}=\{a\}$ so $V/\{0\}$ is essentially just the points of $V$ itself.

A one dimensional subspace of $\mathbb{R}^n$ is a line through the origin. The corresponding affine subspaces are all lines (not just those through the origin). Specifically, if $U$ is a line through the origin, then $a+U$ is a line parallel to $U$ which passes through $a$.

Likewise, two dimensional subspaces of $\mathbb{R}^n$ are planes through the origin whereas the two dimensional affine subspaces are arbitrary planes.

The definition you cite is incorrect (so yes, there is an error). Indeed, letting $U = V$ every subset is an affine subspace according to this definition.