Intuition behind the formula for multiple linear regression coefficients from Econometric Analysis Greene 
I was looking into the maths why coefficients change with the addition of new variables and so looked up the formula for multiple linear regression coefficients. This is what I found from section 3.2.2 in Greene.
In comparison to the simple linear regression versions, I really don't know how to interpret these as I don't see any intuition behind these formulas. The denominator looks a bit like the covriance for t and g which makes a bit of sense to me. I really don't understand what the numerator is meant to be though.
Note: the uncapitalised letters represent the variables deviation from it's mean. So $t_i = (T_i-\bar{T})$.
Any help with the intuition of these would be highly appreciated.

I am failing to see anything unnecessarily intricate here.
Let me paraphrase: Greene first regressed Real Investment $(\mathrm Y) $ on a constant $({1}), $ time Trend $(\mathrm T) $ and Real GDP $(\mathrm {G}). $
The normal equations are:
$$ \begin{bmatrix}n & \sum T_i & \sum G_i\\ \sum T_i & \sum T_i^2 &\sum T_iG_i\\ \sum G_i & \sum T_iG_i & \sum G_i^2\end{bmatrix}\begin{bmatrix}b_1\\b_2\\b_3\end{bmatrix} = \begin{bmatrix}\sum Y_i\\ \sum T_iY_i\\ \sum G_iY_i\end{bmatrix}.\tag 1$$
From the very first equation, it can be readily seen
$$b_1 = \bar Y-b_2\bar T -b_3\bar G\tag 2.$$
Using $(2) ,$ now
$$\begin{bmatrix}\sum t_i^2 & \sum t_ig_i\\ \sum t_ig_i & \sum g_i^2\end{bmatrix}\begin{bmatrix}b_2\\b_3\end{bmatrix}= \begin{bmatrix}\sum t_iy_i\\ \sum g_iy_i\end{bmatrix}; \tag 3$$
Now the $\rm OLS$ estimates, $b_2,~ b_3$ are:
\begin{align}b_2&= \frac{\sum_i t_iy_i\sum_i g_i^2-\sum_i g_iy_i\sum_i t_ig_i}{\sum_i t_i^2\sum_ig_i^2 -(\sum_ig_it_i)^2},\tag 4\\ b_3&= \frac{\sum_i g_iy_i\sum_i t_i^2-\sum_i t_iy_i\sum_i t_ig_i}{\sum_i t_i^2\sum_ig_i^2 -(\sum_ig_it_i)^2}.\tag 5\end{align}
Now comes the important part that OP was seeking. Greene investigated what would happen if $\rm T$ is omitted. He computed $$b_\mathrm{YG} = \frac{\sum_i g_iy_i}{\sum_ig_i^2}= 0.00533.\tag 6$$
Now, dividing both numerator and denominator in $(5) $ by $\sum_i t_i^2\sum_ig_i^2$ and using $r^2_\mathrm{GT}:= \frac{(\sum_ig_it_i)^2}{\left(\sum_i t_i^2\sum_ig_i^2\right)}, $
$$ b_\mathrm{YG|T}= \frac{b_\mathrm{YG}}{1-r^2_\mathrm{GT}}-\frac{b_\mathrm{YT}b_\mathrm{TG}}{1-r^2_\mathrm{GT}}= 0.1080157,\tag 7$$ which is over $\approx 20.2656$ times $b_\mathrm{YG}.$
Via this example, Greene concluded that for a three-variable regression with a constant, the relation between the regression coefficient when the variable is omitted and when it is present as $$b_\mathrm{Y2|3}= \frac{b_\mathrm{Y2}-b_\mathrm{Y3}b_{32}}{1-r^2_{23}}.\tag 8$$

Reference:
$[\rm I]$ Econometric Analysis, William H. Greene, 2018, Pearson Education; chapter $3, $ pp. $30-32.$
