So $x_1(t)$ and $x_2(t)$ are equal up to a certain time $t_0$, and diverge after that time. What this definition is saying is that, up to time $t_0$, the system will produce the exact same output, without regard for whether the input is actually $x_1(t)$ or $x_2(t)$.
If the system is non-causal, then it may "know" whether the input is actually $x_1(t)$ or $x_2(t)$, because it can "look ahead" in time beyond $t_0$, and it could modify its output accordingly, even before $t_0$.
As an example, consider a lightbulb connected to a battery and to a switch. The switch is initially on, and the lightbulb is also on. At time $t=0$, you decide whether you'll turn off the switch at time $t=60$ (one minute). The lightbulb, being causal, cannot know in advance what will happen, and will stay on at least up to $t=60$. A non-causal lightbulb will know in advance what will happen and may turn itself off at, say, $t=30<60$.
I hope this makes sense -- let me know if you need more clarification.