Suppose $X$ is a stochastic process with a certain probability distribution that is not time-dependent. $X$'s value is assumed to be a real number.
Now we want to take the average of $X$ over every possible time paths. So for example, if $X$ is normally distributed with mean 0, one path has: at time $t=0$, $X$ comes out to be 1, at time $t = 0+\Delta t$, X comes out to be -2, and so on. The other path has: at time $t=0$, $X$ comes out to be -3, at $\Delta t$ $X$ comes out as -6, etc. So we integrate over every path and calculate the average of the value of $X$.
How do we do this?
For each fixed $\omega \in \Omega$, the mapping $t \mapsto X_t(\omega)=:f(t)$ is a (deterministic) function. The average of a function $f$ over a time interval $[0,T]$ equals
$$\frac{1}{T} \int_0^T f(s) \, ds.$$
Consequently,
$$\frac{1}{T} \int_0^T X_s(\omega) \, ds$$
is the average over $[0,T]$. Since, by assumption, the distribution does not depend on the time, we have in particular
$$\mathbb{E} \left( \frac{1}{T} \int_0^T X_s(\omega) \right) = \mathbb{E}(X_0),$$
i.e. the average over time and space equals the expectation value of the (fixed) distribution of our process.