Conceptual Intuition of the CLT

As we know, the CLT says the standardized sample mean (or equivalently, the sample sum) converges (in distribution) to the standard normal. Now, the sample mean is random (as it depends on the sample) therefore it has a distribution (called the sampling distribution) which, for a fixed $n$, is constructed by taking many samples of size $n$, calculating the mean for each sample, and then drawing the histogram. The CLT says that if we keep increasing $n$, the sampling distribution of the mean will converge (in distribution) to the standard normal distribution.