The Central Limit Theorem (CLT) applies to any probability distribution, provided that it has finite variance.
In everyday practical statistics, however, the CLT is often (almost always?) invoked in situations where the "underlying distribution" is unknown.
I see no a priori reason to expect that such an unknown distribution will have finite variance. (In particular, see this answer.) Therefore, I see no a priori justification for such extensive reliance on the CLT.
What's wrong with my reasoning here?
Importantly, methods that are based on invalid assumptions of normality can result in spectacular failures (resulting for underestimation of the probability of rare events).
One should not seek an a priori reason; rather the reason should be sought in the nature of each case separately. Suppose, for example, that you're talking about weights of hamsters. It is reasonable in practice to assume that the distribution of those weights not only has finite variance but is of such a form that convergence in the CLT will not be slow.