What is the use of Spectral Theorem?

What you describe is more or less the spectral theorem for normal operators on Hilbert space, which in turn yields the Borel functional calculus for such operators. See Chapter 12 of Rudin's Functional Analysis (2nd ed.), in particular 12.21-12.24.

As to why to Borel FC might be useful: well, it gives us a way of using what we know about bounded measurable functions to prove and construct things about (normal) operators on Hilbert space. For instance, the polar decomposition of a bounded operator $T$ on Hilbert space can only be proved, as far as I know, by using Borel functional calculus for the operator $T^*T$. In turn, the polar decomposition gives probably the quickest way to show that the unitary group of $B(H)$ is connected in the norm topology.

Note also that if $T$ is normal and belongs to a von Neumann algebra $M\subseteq B(H)$, then so does $f(T)$ for $f$ a bounded Borel function on $\sigma(T)$. So the functional calculus is - and needs to be - used when developing the structure theory of von Neumann algebras, because it allows us to construct elements with prescribed properties that still lie inside the specified von Neumann algebra.