Why is $R^2$ the proportion of total variance of the data explained by the model? I have that $R^{2} = 1 - \frac{\text{RSS}}{\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{2}}$.
Also, $\text{RSS}= {\sum_{i=1}^{n}(Y_{i}-\hat{Y_{i}})^{2}}$ for the simplest linear model with only the intercept term.
I also know that $\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{2}$ is the total variance for the intercept only model and that $\frac{\text{RSS}}{\frac{1}{n}{\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{2}}}$ is approximately $\frac{\text{var. of model}}{\text{variance}}$. 
However I still don't get why $R^{2}$ is the proportion of total variance of the data explained by the model.
 A: There is an error in your equations, $RSS = \sum(Y_i - \hat{Y}_i)^2$
Maybe it would help not looking at so many equations to understand. 
RSS is the sum of the residual variance, basically the sum of all the variance that the model can't explain.
Therefore
$\frac{RSS}{\sum{(Y_i - \bar{Y})^2}}$ is $\frac{unexplained \ variance}{Sum  \ of \ all \ variance}$
so
$1- \frac{unexplained \ variance}{Sum  \ of \ all \ variance} = \frac{Sum  \ of \ all \ variance - unexplained \ variance}{Sum  \ of \ all \ variance} = \frac{explained \ variance}{Sum  \ of \ all \ variance} $
Does this help?
A: We have $TSS = \sum_i (Y_i - \bar{Y})^2,\  RSS = \sum_i(Y_i - \hat{Y}_i)^2,\  ESS = \sum_i(\hat{Y}_i - \bar{Y})^2$
$TSS$ - total variance, $RSS$ - residual variance, $ESS$ - regression variance
From ANOVA identity we know that
$$TSS = RSS + ESS$$
So we have $R^2 = 1 - \frac{RSS}{TSS} = \frac{ESS}{TSS}$. From last equation you can clearly see that $R^2$ states how much "variance" is explained by the regression
