Trouble understanding the Central Limit Theorem's real life application As I understand it, the CLT states that for a sufficiently large sample size "n", the sampling distribution of the mean from a given population will approximate a normal distribution. Also as the sample size grows, the better the mean approximation becomes. However, when watching a lecture about confidence intervals, the lecturer explains that an assumption when creating confidence intervals is that the population distribution is normally distributed, and if it is not "you should use a large enough sample and let the CLT do the normalization magic for you"; and then proceeds to work off of a single sample. In another video, a different lecturer creates a sampling distribution of the mean and states that we can create a confidence interval from the sampling distribution. So which one is correct and what is the difference. Is the sampling distribution explanation of the CLT only a "proof" of what the theorem dictates and extracting a single large sample is enough? Or am I missing something?

The central limit theorem as it's name says is a assymptotic result.
It lays out conditions under which we can conclude some very useful properties about properly normalized sums of a increasing set of random variables. A version of the CLT has the following form,
$$\frac{S_n}{s_n}\to^D N(0,1)$$
where $S_n$ is the partial sum of zero-mean random variables (WLOG as we can always subtract the mean if it's finite) and $s_n$ is a normalizing sequence. Under the assumption of independent and identically distributed random variables with mean $\mu$ and standard deviation $\sigma$ we get the common form,
$$ \frac{\sqrt{n}\left(\sum_{k=1}^nX_k - \mu\right)}{\sigma} \to^D N(0,1)$$
Notice that when the sample size is large enough, we get a good pivotal quantity to build confidence intervals, also by the interval-test duality we also get a way to test hypothesis! Search a little about the Wald's test, for example.
There are many other CLTs, some don't assume indepence, but only that the random variables are a Martingale sequence,
$$\mathbb{E}[X_n|\mathcal{F}_{m}]= X_m, \; 1 \leq m < n$$
thus the CLT has many forms and extensions to different sets of assumptions.
It's a beautiful theorem that gives us tools to do inference when working with large sample sizes.
