Recall that in a regression setting, the F statistic is expressed in the following way.
$$
F = \frac{(TSS - RSS)/(p-1)}{RSS/(n-p)}
$$
where TSS = total sum of squares and RSS = residual sum of squares, $p$ is the number of predictors (including the constant) and $n$ is the number of observations. This statistic has an $F$ distribution with degrees of freedom $p-1$ and $n-p$.
Also recall that
$$
R^2 = 1 - \frac{RSS}{TSS} = \frac{TSS - RSS}{TSS}
$$
simple algebra will tell you that
$$
R^2 = 1 - (1 + F \cdot \frac{p-1}{n-p})^{-1}
$$
where F is the F statistic from above.
This is the theoretical relationship between the F statistic (or the F test) and $R^2$.
The practical interpretation is that a bigger $R^2$ lead to high values of F, so if $R^2$ is big (which means that a linear model fits the data well), then the corresponding F statistic should be large, which means that that there should be strong evidence that at least some of the coefficients are non-zero.