This is my understanding of what the Central Limit Theorem (CLT) is: if you take a number of samples, each containing a large number of observations, and calculate their respective sample means, then these sample means will have an approximately normal distribution, regardless of the parent population's distribution.
I was told that the CLT can be used to explain why a Binomial distribution is approximately normal for large values of n, but do not see why this is the case.
What the central limit theorem actually says (there are several CLTs, so let's discuss a version of the classical CLT for concreteness) doesn't describe what happens at "large n" but in the limit as $n\to\infty$; it doesn't apply regardless ofthe distribution, but under a few conditions (the mean and variance of the distribution must be finite for example); and it doesn't describe what happens to the distribution of sample means but to the distribution of standardized sample means ($Z_n=\frac{\bar{X}-\mu_X}{\sigma_X/\sqrt{n}}$) - that is, subtract the population mean of the distribution of $\bar{X}$ and divide that difference by the standard deviation of the distribution of $\bar{X}$.
Note that if you multiply numerator and denominator by $n$ you get
$$Z_n=\frac{\sum_i{X_i}-n\mu_X}{\sigma_X\sqrt{n}}$$
Now $n\mu_X$ is the mean of the distribution of $\sum_i{X_i}$ and $\sigma_X\sqrt{n}$ is the standard deviation of the distribution of $\sum_i{X_i}$, so that same statistic $Z_n$ is also a standardized sum of variables.
So in the limit the distribution function (i.e. the cdf) of a standardized sum also converges to a standard normal distribution function under the same conditions (it's actually the same object).
We might anticipate then that for finite $n$ that as $n$ increases the distribution function of the standardized sum more closely approaches the standard normal distribution function.
In fact if you look at the Berry-Esseen theorem, this is actually the case (though it's a little more restrictive than the CLT) -- you have bounds on how far the two cdfs can be apart, and these bounds decrease as $n$ increases.
The hand-wavy version
The mean is just a constant times the sum, so if the distribution-function of the mean looks close to normal, the distribution-function of the sum (which only differs in the scale on the x-axis) also "looks close to normal".
The left hand plot is the distribution function of the mean of $n$ Bernoulli variates; the right hand plot is the distribution function of the sum (i.e. of a binomial). They have the same shape but are on different scales. If you standardize either variate you change the scale again (but not the shape); its the distribution of the standardized variates that are shown to converge in the limit under the CLT.
