Well, you asked for a "simpler" description for the regression in the context of least-squares-fit. I find this a simpler one, although the basic ingredience might be more difficult than the usual approach: thinking in terms of n dimensions, where n is the number of observations, i.e. questioned people or in your case n=5 by the given five points.
If this can be imagined without too much hassle, then the following should also be a "simpler" explanation.
The variables are then vectors from the origin into that n-dimensional space, and we have from your example the points (1,3)(2,8), and extend this to for instance (4,5)(5,7)(3,6) to have n=5 points.
Then we can rewrite this as the vector X pointing to (1,2,4,5,3) in that five dimensions and Y pointing to (3,8,5,7,6).
Also we define the "mean-vector" $\small M(k) $ pointing to any coordinate $\small k \cdot (1,1,1,1,1)$ on the (multidimensional) diagonal.
Clearly $\small M((1+2+4+5+3)/5)=M(3)=3 \cdot (1,1,1,1,1) $ is the vector pointing to the mean of the coordinates of X and $\small M((3+8+5+7+6)/5)=M(29/5)=5.8 \cdot (1,1,1,1,1) $ is the vector pointing to the mean of the coordinates of Y.
Note, that by this the point M(3) is the point on the diagonal which is nearest to the tip of X : but to express the distance of two points we simply need the pythagorean formula with their coordinates - and come up exactly with the least-squares-criterion. This is also true for the distance of M(5.8) from the tip of Y . Just as a spinoff of that model we "see" immediately that the "means" are the best approximates to a set of values in the sense of least-squares.
The same model can now be used to express the linear regression: we want to compose the mean M(.) and the X-vector in such a way that we come nearest to the tip of Y. Or said differently, we want to find a vector $\small \hat Y = a \cdot M(3)+b \cdot X = M(3 a)+b \cdot X $ with a and b to be found such that the tip of $\small \hat Y $ is nearest to the tip of Y.
That's linear regression in the sense of least-squares-approximation.