Definition of an affine set

Note that the second definition is a generalisation of the first. A set is affine iff it contains all lines through any two points in the set (hence, as a trivial case, a set containing a single point is affine).

(Thanks to @McFry who caught a little sloppiness in my original answer.)

Use induction: Suppose it is true for any collection of $k \le n-1$ points (it is trivially true for $n=1$) and consider the point $\sum_k a_k x_k$.

If $a_k = 1$ for all $k$, we must have $n=1$ (since $\sum _k a_k = 1$), so we are finished. Hence we can assume that $a_k \neq 1$ for some $k$. By renumbering, we can assume that $a_n \neq 1$.

So, suppose $a_n \neq 1$, then you can write $\sum_k a_k x_k = \sum_{k<n}a_k x_k + a_n x_n = (1-a_n) \sum_{k<n}{a_k \over 1-a_n}x_k + a_n x_n$, and note that $\sum_{k<n}{a_k \over 1-a_n} = 1$.

The other direction is immediate.

Given points $x_1, \ldots, x_n$, you have by assumption all the lines from $x_i$ to $x_j$ contained in your space. Now you could argue that the inside of the shape is just lines between points on the lines that form the edges.