On page $3$ of these slides, it says:
It’s often a good approximation to make the random force “white” and “uncorrelated in time”.
And this equation just pops out of nowhere.
I know white noise, but I have no idea what this equation means. Where does this definition come from? And what does it mean?
This equation is the autocorrelation, see wikipedia . (or autocovariance , depending of usages :-) . Quite the same by a $\sigma$ ).
Not correlated in time is like each sample being independently distributed in time. So the correlation between $x_{t0}$ and $x_{t1}$ is 1 if $t0=t1$ otherwise 0.