I will not go too deep into the details but here some information to facilitate your understanding of these ubiquitous statistics in biology.
The F, $r^2$ and p-value are all equivalent statistics to measures the goodness-of-fit of a model against the data.
The F is the ratio of the $unexplained~variance~/~explained~variance$. I let you look out what the variance is. The F (originating from a F-test) allows, using the proper F-distribution cumulative density function (cdf) for the number of observation you have, to compute a p-value.
The p-value gives you the probability of obtaining the observed fit with your model (here a linear one) by random chance only. For example in your case a p-value=0.01 means that you have a 1% chance to observe this correlation just by chance. Usually the p-value threshold used to claim statistical significance is 0.05 (i.e. 5% chances to observe that fitting by random chance). In linear regression, the lowest the p-value the closest the distance (called variance) of the observed data point to the line.
The $r^2$ is the coefficient of determination (or the square of the correlation coefficient). This measure is very interesting as it give you an estimate of the "proportion of the data" explained by your model. For example a $r^2$=1 means that 100% of the variance of the data is explained by your model, in other terms your model perfectly fits the data. A $r^2$=0 in the other hand means that your model does not at all explain the data (no fits between your model and the data).
The $r^2$ and the p-value can be computed for any type of fitting, like trying to fit an exponential curve instead of a linear one. What you usually look for when modeling data is the highest $r^2$ possible. As all of this statistics are linked a higher $r^2$ means a lower p-value for an identical number of observations.