Application of Kronecker Weber Theorem

A theorem like that of Kronecker and Weber is not measured in terms of applications, it is measured in terms of insight and the potential to generate powerful generalizations. It has given rise to Kronecker's theory of complex multiplications and to one of Hilbert's 23 problems, and is a guiding theorem for classical class field theory. It has various proofs, each of which uses an important technique (Lagrange resolvents and Stickelberger, Hilbert's abelian crossings, decomposition and ramification groups, local fields and a lot more).

To amplify the answer of @Piquito, any time you want an example of an abelian extension of $\Bbb Q$, you can use KW and find one inside a suitable cyclotomic extension.

Want a cyclic extension of degree $3$? There’s one inside each $\Bbb Q(\zeta_n)$ for $n=9$ and any prime $n\equiv1\pmod3$. Want more than one cubic extension inside some field $k\supset\Bbb Q$? Take two or more of the fields mentioned above, and form compositum. For instance $\Bbb Q(\zeta_7,\zeta_9)=\Bbb Q(\zeta_{63})$ has in it four cyclic cubic extensions of $\Bbb Q$.

Of course the real force of KW is that this is the only way to get finite abelian extensions of $\Bbb Q$.

It is one of the most important theorems that exist and, in particular, is useful to characterize Abelian algebraic extensions i.e. those whose Galois group is commutative. In the difficult Class Field Theory, cultivated by few persons, this theorem has a special relief.