Suppose we have a population in which we are interested. Imagine we pick an individual from the population completely at random, with all individuals equally likely to be chosen. Then the population is just the sample space of an experiment where the probability of each outcome is $1/N$, where $N$ is the size of the population.
Suppose also we have a measurement we can do for each individual of the population. If, whenever we do our experiment of picking an individual, we then take our measurement, the result of the experiment is just a random variable. Let's call that RV $X$.
Let's take the average value of $X$ for all individuals in the population. That's a number — which we will write $\mu$ and call the population mean of $X$ — which is a number computed by looking at the whole population. In other words, $\mu$ is a population parameter.
We could also take the standard deviation of all of the $X$ values for all the individuals of the population — which we will write $\sigma$ and call the population standard deviation of $X$ — which is then another population parameter.
If instead we had limited resources, we could not measure $X$ for every individual in the population and so we could not actually compute $\mu$. Instead, for some sample size $n$ which is less than $N$ (so: $n<N$), we might be able to pick a sample of size $n$ of individuals from the population. For this sample, we can compute the sample mean $\overline{X}$
Actually, we can think of this process of picking samples of size $n$ from our target population as another random experiment, call it the sampling experiement. The sample space of this new experiment would be the set of all collections of $n$ individuals from our original population. Picking a random sample of size $n$ from our target population is now just doing the sample experiment once and looking a the outcome.
Every time we do the sampling experiment, we can do the measurement in which we are interested for the individuals in that sample, and compute the sample average $\overline{X}$. This is a number associated to the outcomes of an experiment — in other words, $\overline{X}$ is a random variable for the sampling experiement.
If the original population was finite (so: $N<\infty$), then the sample space for the sampling experiment — the set of all samples of size $n$ out of our original population — is finite too (although probably very large). So if we had the probabilities for the outcomes of the sampling experiment, we could write down the probability distribution of this discrete RV, $\overline{X}$.
Suppose we have a mechanism which chooses all samples of size $n$ with equal probability. In other words, suppose the probabilities of all outcomes of the sampling experiment are the same. If so, then we say that any chosen sample is a simple random sample [SRS].
[Note: this is actually quite a strange definition. If someone gives you a sample and tells you it is an SRS, you cannot tell if that is correct simply by looking at that sample. Instead, you have to look at the mechanism they used to choose the sample, and see if it would have had the same chance of producing any sample of size $n$. If so, that was an SRS.For the rest of this course, we will always assume that whenever we can, we will choose an SRS. This means that we have an experiment, the sampling experiment, and probabilities (all equal) for each of the outcomes (which outcomes are samples of size $n$ from the original population), and an RV $\overline{X}$ for that experiment. We call the probability distribution of this RV the sampling distribution of $\overline{X}$.Note also that having an SRS does not mean that every individual was equally likely to be chosen to be in the sample; it means that every sample of size $n$ was equally likely to be chosen. This is quite hard, there are even examples of sampling processes which choose each individual with equal chance ... but they do not choose each sample of size $n$ with equal chance, so they are not actually SRSs.]
The main content of SIS §6.1 is that $\mu_\overline{X} = \mu$ and $\sigma_\overline{X} = \frac{\sigma}{\sqrt{n}},$ where $\mu_\overline{X}$ and $\sigma_\overline{X}$ are the mean and standard deviation of the RV $\overline{X}$ for the sampling experiment; that is, $\mu_\overline{X}$ and $\sigma_\overline{X}$ are the mean and standard deviation of $\overline{X}$ as we look at all samples of size $n$.
What this is saying is that
Unfortunately, SIS §6.1 does not mention SRSs or the requirement that we are looking at SRSs when we write down the above formulæ for $\mu_\overline{X}$ and $\sigma_\overline{X}$. Nevertheless, this is an important hypothesis. In fact, the hypothesis that we must work with SRSs is also important in the Central Limit Theorem and the other results in SIS §6.2.