The answer is that it only does so when three properties are met by the population from which the sample is drawn:
1) the population has a central tendency, that is, a propensity for values to be near the mathematical average for the population as a whole
2) the departures from average are symmetric, that is, there is the same chance for an individual to have a value greater than average as for it to have a value less than average
3) the probability of a given departure from average decreases rapidly with its size - small departures are very likely, larger ones are less likely, very large ones are extremely unlikely
If any one or more of these is not true, the distribution will not be normal. However, for extremely large samples, most real world data approximately meet these conditions anyway, even if they do not for sample sizes of 30.
In reality, small samples from normally distributed populations are more likely to fit the Student-T distribution, which is like normal but more sharply pointed.
n>30 is not inherent in the Central Limit Theorem, it is a convention that makes the proof calculations easier. It facilitates the conversion to polar coordinates and back that is used to prove that the Gaussian formula is a valid probability distribution