Some thoughts and links on the analysts' NCG, from someone who doesn't practice it. Caveat lector. (Some edits made to erroneous history.)
NCG a la Connes was originally non-commutative differential geometry (which is why extra structure is needed in, say, the definition of spectral triple). Having only recently looked at Connes' original long, two-part paper in the Publications of the IHES, I think this is a better place to start than the later Big-Picture-Which-Is-Really Fundable works of various people thereafter. Work of Connes and & Moscovici and others on trying to generalize the Atiyah-Singer index theorem also give some indication of the original motivation. (This is where someone more expert than me should really step in and say something about the work of
C & M Mischenko, Kasparov and their co-authors on the Novikov conjecture, or indeed the work of Higson et al. on the Baum-Connes conjecture.)
It was only afterwards that Connes started championing an NCG perspective on The Standard Model. (Although, if you want real connections with mathematical physics, there was some work of Jean Belissard on identifying gaps in spectra of certain operators in a quantum-mechanical model with K-theoretic invariants of associated C*-algebras. See this paper of Kaminker and Putnam and the references therein for more details.)
Personally, I am bit leery of the Big Picture motivation for noncommutative geometry, at least of this variety. The most useful variant of such motivation that I can think of, is that degenerate group actions on topological spaces give rise to better-behaved homotopy groupoids; thanks to a theorem of Rieffel (IIRC), when the group action on the space is nice, the commutative C*-algebra of the quotient space is Morita equivalent -- i.e. has "the same module theory" -- as the noncommutative C*-algebra of the homotopy groupoid.
I apologize for the rambly nature of this answer, but with all due respect to Anweshi I think his question, or at least the version of it which I can currently see, is so broad (as per Anton's original comments) that the only responses are either encyclopaedic - I haven't even had space to mention the historical role played by Brown-Douglas-Fillmore theory, for instance - or sales pitches. Nevertheless, if someone can persuade Nigel Higson to drop by, I'm sure he could give a much better answer ;)