This is most easily understood if you have some experience with probability theory. Mathematically speaking, a random variable is a measurable function from some background probability space into another space. It is just a function. Say, we're measuring the height of the next person passing by my window. We could define a stochastic variable from the background space into the real numbers and use it to describe this height. Now we just have a random variable. When somebody actually passes by, we would have a real number. An realization of that random variable. Let's say that realization is 1.9 m. It could have been infinitely many other numbers.
Your situation is the same. You have a background probability space and a function from that space into some other space. This time we call the function a stochastic process (because it is a function into some space indexed by time, for instance), and the realization of the stochastic process is now called an observed time series. Apart from the names, the situation is as above.
So to answer your questions more directly, think of the ensemble as the set of all the (observed) time series you could possibly see as realizations of a stochastic process, the one you actually observe is just one of them. This is why every member of the ensemble is a possible realization of the process.
The background space is a mathematical construction to define random variables. Another way to think of it would be to say that if we could repeat our observation, we would possibly have seen another time series. Hence, the same stochastic process can give rise to multiple realizations (observed time series and members of the ensemble). The observed time series (members of the ensemble) are the numbers you actually observe, while the stochastic process is a mathematical construction, explaining where the numbers came from. We are often interested in gaining knowledge on the stochastic process but we only have the information in the observed time series.