The intuition behind it is that when we take batches of samples from some arbitrarily shaped distribution, and we summarize the information by looking at the mean values of the batches of samples, we find that those mean values are moving away from the arbitrarily shaped distribution. The larger are the batches, the more those means approach a normal distribution.

In other words, the means of large batches of samples from some funny shaped distribution themselves constitute a sequence of numbers, and that sequence follow a normal distribution. Or closer and closer to one the larger the batches are.

This observation legitimizes our uses of statistical inference tools derived from the normal distribution, like confidence intervals, provided we are working with large enough batches of samples.