I am new to affine space, I looked through the wikipedia page, and have problem understanding the statements below.
Can anyone try to help explain in more detail?
I'm not an expert on affine spaces either, but this is how I understand it.
Suppose you have two different points $p_1 = (x_1,\ y_1)$ and $p_2 = (x_2,\ y_2)$. Then you can make lineair combinations $\alpha p_1 + \beta p_2$ of those points. If you restrict the choices for $\alpha$ and $\beta$ to those values for which $\alpha+\beta=1$, then you're making affine combinations. Together they make the line that goes through $p_1$ and $p_2$. You can see this by writing $\beta=1-\alpha$, so that your combination is given by $\alpha p_1 + (1-\alpha)p_2 = p_2 + \alpha(p_1-p_2)$. Starting from $p_2$ you can choose any factor $\alpha$ that multiplies the difference between your two points. This is exactly the line that goes through those points.
The affine hull is the set of all affine combinations of the points in your set. So, if you have only two points this is the line that goes through these points, as explained above. If you have three points, you get two lines: the line through $p_1$ and $p_2$ and through $p_1$ and $p_3$. These two lines (assuming they are different) span a plane that goes through $p_1$, $p_2$ and $p_3$ (you can also use the line through $p_2$ and $p_3$, but it will lead to the same plane).
You can see that two lines span a plan by drawing two lines on a piece of paper. If you imagine your paper is infinitely large, the plane spanned by your lines is the paper on which you drew the lines.
Similarly, with four points you get three lines which (assuming they are different) span a space that goes `through' all four points. This is $\mathbb{R}^3$.
