Recall that the commutative Gelfand-Naimark theorem, commutative C*-algebras are precisely the C*-algebras $C(X)$ of continuous complex-valued functions on compact Hausdorff spaces $X$, namely their spectra. So invariants of $C(X)$ are telling us something about the spaces $X$. What does K-theory tell us in particular?
By Swan's theorem, finitely generated projective $C(X)$-modules are the same thing as vector bundles over $X$, so $K_0(C(X))$ turns out to be precisely the topological (complex) K-theory $K^0(X)$ of $X$, and similarly $K_1(C(X))$ turns out to be precisely $K^1(X)$.
Topological K-theory is historically the first example of a generalized cohomology theory, and as such it contains information about spaces $X$ similar to the information given by singular cohomology. In fact the Chern character isomorphism implies that
$K^0(X) \otimes \mathbb{Q} \cong \prod_n H^{2n}(X, \mathbb{Q})$ $K^1(X) \otimes \mathbb{Q} \cong \prod_n H^{2n+1}(X, \mathbb{Q})$
so, after tensoring with $\mathbb{Q}$, topological K-theory encodes roughly the same information as ordinary cohomology, except that it forgets the $\mathbb{Z}$-grading, retaining only a $\mathbb{Z}_2$-grading. With stronger tools like the Atiyah-Hirzebruch spectral sequence you can attempt to compute $K^0(X)$ itself by relating it to cohomology as well.
Hence for commutative C*-algebras, K-theory gives us information similar to the information of the singular cohomology of the spectrum. For noncommutative C*-algebras, K-theory gives us a way to understand the "algebraic topology" of the "noncommutative spaces" of which those C*-algebras are (heuristically) the spaces of functions. This idea goes by the name of noncommutative topology.