Motivation Of Correlation Coefficient Formula Definitions
correlation coefficient $= r = \frac{\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=1}^{n}(x_i - \bar{x})^2\sum_{i=1}^{n}(y_i - \bar{y})^2}}$
My Question
What is the motivation of this formula? It's supposed to measure linear relationships on bivariate data, but I don't understand why it would do that as defined. For example, Riemann integrals are said to measure area under a curve, and that makes sense because $\sum f(x_i)\Delta x$ is adding areas of rectangles under the curve $f(x)$ approximating its area more and more as we take more samples. Does such an intuition exist for the correlation coefficient? What is it? My background in statistics is nothing but a bit of discrete probability. I know histograms, data plots, mean, median, range, variance, standard deviation, box plots and scatter plots at this point (from reading the first weeks material on an introductory statistics class). 
My Research
All of the "Questions that may already have your answer" seemed to either be asking about what the formula said mathematically or asked questions that were more advanced than my knowledge.

Suppose we have a scatterplot of heights X and weights Y of n subjects.
The 'center of the data cloud' is at the point $(\bar X,\,\bar Y)$.
One might expect a positive association between heights and weights.
Points above and to the right of the center make a positive
contribution to the sum $\sum (X_i -\bar X)(Y_i - \bar Y).$
So also do points below and to the left of center.
Points that might suggest a negative association will be above and
to the left of center or below and to the right of center. For them,
the product $(X_i -\bar X)(Y_i - \bar Y)$ will have a negative
and a positive factor, thus a negative product. So such points
will make a negative contribution to  $\sum (X_i -\bar X)(Y_i - \bar Y).$
The denominator is essentially the product of the numerators of the
standard deviations of X and Y. The effect of the denominator is to
make $r$ a quantity without units. In the US system of measurements
the numerator has units 'foot-pounds', and the denominator has the
same units, so $r$ has no units. If the subjects were weighed and measured
in the metric system, the correlation of their weights and heights would
be numerically the same as if they were weighed and measured in the US
system. 
Also, inclusion of the denominator scales correlations $r$ so that
they lie between $-1$ and $+1,$ where $r = 1$ means the points
perfectly fit an upward sloping line (regardless of the numerical
value of the slope, which has units), and $r = -1$ means the points
perfectly fit a downward sloping line.
If either the SD of the X's or the SD of the Y's is 0, then the points
lie on either a vertical line or a horizontal line, respectively. In
either case the denominator of $r$ would be $0$ and the correlation
is not defined.

In the plot below, there is a strong linear component to the positive
association of X and Y: $r = 0.968.$ The horizontal and vertical grid
lines cross at $(\bar X, \bar Y).$ Each of the dark green points makes a positive
contribution to the numerator of $r,$ as discussed above. Also, the
two red points make (slight) negative contributions to the numerator of $r.$

