In the definition below, I am unsure on what the $[\omega]$ in $X(t)[\omega]$ is meant to signify.
Is it saying that we first observe a stochastic process, create the mapping with the seen values of the random variables, then we denote values of the random variables by $X(t)[\omega]$?
Your stochastic process $\{X(t),t\geq0\}$ is defined on some underlying set $\Omega$ (which we don't make reference to very often). This way, for each $t\geq0$, $X(t)$ is a function $$X(t):\Omega\to\mathbb{R}.$$ $\omega$ is just a generic element of the underlying space $\Omega$. Thus for each $t\geq0$ and $\omega\in\Omega$, $X(t)(\omega)$ is a real number. By fixing $\omega\in\Omega$ we can think of the mapping $$t\mapsto X(t)(\omega)$$ as a real-valued function over time.
The definition provided is just saying that the set of all $\omega\in\Omega$ for which the above mapping is continuous is required to have probability measure 1.