First, recall the slogan:
Small constructions are good for making calculations, but large constructions are good for proving theorems.
K-theory is certainly a large construction.
In general, K-theory seems to turn up in topology when the following slogan holds:
Chain compex good; homology bad.
You can often construct exactly the same invariant using K-theory or without, but K-theory makes the extra structure in the chain complex visible (such as Poincare duality), which makes it possible to prove theorems. As a low-dimensional topologist, the example I have in mind is Ranicki's symmetric signature. More basically, the key observation in Milnor's proof that the Alexander polynomial is palindromic is that if you construct it from the chain complex by using Reidemeister torsion, the Poincare duality becomes evident. Similarly, the Blanchfield pairing (linking pairing on the infinite cyclic cover of a knot complement) can be constructed as the symmetric signature (L-theory), which lets you see Poincare duality.
This seems completely typical. You can squish everything into the middle dimension and get information (Wall's finiteness obstruction; Alexander polynomial; Branchfield pairing...) or you can use a large K-theoretic construction to get the same information in a way which preserves the chain complex context from which it comes, revealing properties which come from the grading.