Confusion can arise by calling the unknowns $x$ and $y$. In the picture on the left, they are actual $x,y$ coordinates of a point with geometrical significance. In the picture on the right, $x,y$ are called $x$ and $y$, but they're not connected to $x$ and $y$ coordinates anymore. I think it's conceptually better to separate the left picture and right picture completely, and when trying to understand the picture on the right, don't even call the unknowns $x$ and $y$, because they're not going to be $x$ and $y$ coordinates of anything in our second picture.
Take vectors $$u=\begin{pmatrix} p \\ q \end{pmatrix}$$ $$v=\begin{pmatrix}r\\s\end{pmatrix},$$ $$w=\begin{pmatrix} \alpha \\ \beta\end{pmatrix}.$$
Suppose we want to know how to express $w$ as a linear combination of $u,v$. We take unknown constants $a,b$ and ask: How do we achieve $$au+bv=w?$$ This is coordinate-wise equality, so we write out what this means. We want $$au+bv = a\begin{pmatrix} p \\ q \end{pmatrix}+b\begin{pmatrix}r\\ s\end{pmatrix} = \begin{pmatrix}ap+br \\ aq+bs\end{pmatrix},$$ and we want this to be equal to $\begin{pmatrix} \alpha \\ \beta\end{pmatrix}$. So this is equivalent to solving the system of equations $$ap+br=\alpha$$ $$aq+bs=\beta.$$
Instead of calling the coefficients $a,b$, if we call them $x,y$, we can see the relationship you're asking about. However, this translation is a little difficult precisely because we have pre-existing interpretations of what $x,y$ should be. We think of $p,s,\beta$ as the $y$-coordinates of the vectors if we plot them in $\mathbb{R}^2$ (which is what the picture on the right is doing), but this has no connection to the fact that the variables are called $x,y$. The fact that $x,y$ are actually $x,y$ coordinates in some meaningful plot is reflected in the plot on the left in your link.
In general, if ${\bf X}$ is a matrix with columns $x_1,\ldots, x_p$ and if $\beta_1,\ldots, \beta_p$ are numbers, the linear combination $$\sum_{i=1}^p \beta_i x_i$$ is exactly the matrix product $${\bf X}\begin{pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_p\end{pmatrix}.$$ If we denote $$\begin{pmatrix} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_p\end{pmatrix}$$ as $\beta$, then asking whether or not $${\bf X}\beta=w$$ has a solution is equivalent to asking whether $w$ is in the column space of ${\bf X}$, which is the span of the columns of ${\bf X}$
