Direct meaning: The acronym IID means (statistically) independent and identically distributed. Hence, when we say, "Let $X_1, X_2, ..., X_n$ be IID random variables", we are simply saying that we have a set of $n$ random variables that we take to be independent, with the same distribution function (usually this is followed by specifying their common distribution). This type of statistical set-up is called the IID model and it is the most common statistical model in use. It is usual to specify a distributional form for the observable values with some unknown parameters, and the statistical problem then becomes one of estimating the unknown parameters in the distribution from the observed data.
Operational meaning: The above gives the direct meaning of these phrase in terms of setting up a statistical model. However, it is worth understanding where this model comes from in operational terms (i.e., in terms of probabilistic description directly on observable values). The IID model comes from a famous probability theorem called the representation theorem, initially due to de Finetti, but extended by others. For a sequence of exchangeable random variables $X_1, X_2, X_3, ...$ this theorem tells us that these random variables are independent and identically distributed (IID) with distribution equal to their empirical distribution function (conditional on that distribution function):
$$F(x) \equiv \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^n \mathbb{I}(X_i \leqslant x).$$
Operationally, this tells us that the IID model is the result of an assumption of exchangeability of the observable sequence of random variables (i.e., the order-invariance of the sequence). It also tells us that the underlying distribution of these observable random variables is the empirical distribution of the sequence, so any identifiable parameters must be functions of this distribution. If you would like to read a simple explanation of how this theorem applies to frequentist and Bayesian inference, a useful reference is O'Neill (2011).
The sample mean: You question also asks the meaning of the object $\bar{X}_n$. This is the sample mean of the first $n$ data points, which is defined as $\bar{X}_n \equiv \tfrac{1}{n} \sum_{i=1}^n X_i$. The sample mean is a statistic (meaning it is a function of the observed data) and its distribution is determined by the distribution of the underlying random variables (i.e., the empirical distribution of the sequence). In the IID model it is subject to theorems like the laws-of-large-numbers and the central limit theorem (CLT) (under certain additional assumptions).