I may be asking dumb or non-sensical question, but what does the variance of an estimator for a regression parameter (e.g. $\beta_{0}, \beta_{1}$) mean?   How does it even have variance?  Isn't it a constant estimate of a presumed true but unknown constant value?

I have seen a good mathematical derivation of it [here][1] from which I can see for example that 
$$Var(\beta_{1}) = \frac{\sigma^{2}}{\sum{(x_{i} - \overline{x})^{2}}} = \frac{\sum{(y_{i} - \overline{y})^{2}}}{(n-1)\sum{(x_{i} - \overline{x})^{2}}} $$ 
but it is the practical understanding of it that is eluding me.

We don't know from our sample if we are right or wrong, for all we know (even though there is variance in the Y's) we may have come up with the exact true estimators, yet we calculate a variance for them based on variation of sampled X's and Y's from their means .... ???

  [1]: https://stats.stackexchange.com/questions/64195/how-do-i-calculate-the-variance-of-the-ols-estimator-beta-0-conditional-on