What isn't clear from the above is why anyone bothered to define the notion of compactness: in my opinion it is an idea which perfectly embodies the notion of mathematical elegance.
I suppose a deep reason why the notion of compactness is useful comes from the defining axioms of topological spaces: that any finite intersection of open sets is open, but that arbitary intersections are not (eg. consider the intersection of all the open sets (-ε,ε) for all ε between 0 and 1 - the intersection is {0}, which is not open). Given compactness, we can reduce the arbitary intersections to finite ones and then there is no problem.

Here are a few nice consequences of compactness:

  • Proofs get a lot shorter. Compare these two proofs that a continuous image of [ 0,1 ] is closed and bounded:
    • Given any sequence of points (xn) in [ 0,1 ] , it will have a subsequence converging to some point in [ 0,1 ] (because closed and bounded). Knowing this, one can show that the image set also has convergent subsequences to arbitary sequences too: let (yn) be such a sequence in the image set, and then for each yn define a xn such that f(xn)=yn. The given the convergent subsequence of the (xn), can define one for the (yn), which agree in the limit by the continuity of f.
      The subsequence property on the image set, one can show that it is closed and bounded: if not could produce a sequence (yn) such that |yn| > n for every n, which could not have a convergent subsequence. Nasty.
    • [ 0,1 ] is closed and bounded, so compact. The continuous image of a compact set is compact - and hence too closed and bounded. Nice.
    In this way, you can prove many similar things quickly by simply stringing together known lemmas about compact spaces.
  • You can make lots of spaces compact simply by adding an extra point (and making a change to the topology)- eg. adding 'infinity' to the complex plane turns it into the Riemann Sphere. Doing this gives you a sensible way to deal with infinity, and gives rise to the notion of Riemann Surfaces, which are things of great beauty.
  • Its simple formulation and applicability to all topological spaces make it a powerful abstraction mechanism.