Definition
In continuous time, a random time $T$ is a stopping time if for every $t\ge 0$,
$$ \{T\le t\}\in \mathcal F_t. $$Interpretation
By time $t$, you can determine whether the stopping event has already occurred using only information up to time $t$.
Why the definition uses $\{T\le t\}$
In continuous time, events of the form $\{T=t\}$ are usually too small to be useful. The event $\{T\le t\}$ captures whether stopping has happened by time $t$.