What you created here is a model, that tries to reflect reality. But of course, unless we are exceptionally lucky, the model will never reflect reality perfectly. And the standard deviations reflect how confident the model is about itself.
In your question, you stated you generated data with $\beta_0 = 1$ and $\beta_1=2$. Those numbers are the reality your model tries to reflect. Now suppose you didn't tell us those values, just your model. What can we say about your input?
The model tells us the most likely values are $\beta_0 = 1.21042$ and $\beta_1=1.87223$.
But could it be that you that the actual values you put in (the reality) were $1.2$ and $1.9$? Well, therefore we have to look at the standard deviation.
With the given standard deviations, the model tells you it's $68 \%$ sure the true value of $\beta_0$ is in the range $1.09546 - 1.32562$ (minus 1 sd and plus 1 sd). And it's $95 \%$ sure the true value is in the range $0.98038 - 1.4407$ (2 sd away). For $\beta_1$, we can do a similar calculation. That means the numbers $1.2$ and $1.9$ are very reasonable guesses, but that $1$ and $2$ are also not too outlandish.
Now in reality, we often don't have access to the true values of $\beta_0$ and $\beta_1$. We can just take measurements, and make the best model we have. Or sometimes, theorists will come up with a theory that has to be tested on reality, to check whether the model is right or wrong.
As an experimental physicist, you'll run some experiments and maybe get the same values you got. You'll make a model, and can publish this to show that a theory that predicts $\beta_0 = 0$ and $\beta_1=5$ is most definitely wrong (if you can prove your experimental setup is correct). The values you got of $1.21$ and $1.87$ are basically your best guesses as to what the true values could be. But a theory that predicts $\beta_0=1$ and $\beta_1=2$ may well be correct.
Until you come up with an experiment that's more sensitive. Suppose you do the same, and get a model that shows:
\begin{align}
\hat \beta_0 & = 1.19554 \quad \text{with Std. Error} = 0.01279, \\
\hat \beta_1 & = 1.88341 \quad \text{with Std. Error} = 0.02369.
\end{align}
These values allign quite well with your earlier result (showing there was likely no systemic error in your first experiment). But they have much narrower standard deviations, and now also show the theory with $\beta_0=1$ and $\beta_1=2$ is also wrong. But the guesses of $1.2$ and $1.9$ are still holding.