An estimator is a process to construct estimates for a quantity q based on a random sample $X_1$,$X_{2}$, ..., $X_{n}$.
So suppose we want to check how many hours of tv are being watched by people in the US. I can't take into consideration the whole population of millions of people. Instead I consider $1000$ of them. So $X_1,X_2, ..., X_{1000}$ is denoting the number of hours for each of the $1000$ people. And why are these variables? Because $X_1$ can take any value from $0-24$, we don't know for sure. Is this realization correct?
Also please feel free to share any point that a person can miss when reading this kind of thing.
Yes, you are correct. Variables can take on different values for repeats of the same situation. A single random sample is the values of a set of "draws" from a population.
In probability we reason from a distribution to data; in statistics we reason from data to a distribution or model. We want to know about the full population, but without a census, that is not possible. So we take our "best guess" which is found using statistics.