I'm not attempting to completely answer your question, but add my 2 cents.
When doing condensed matter (statistical) physics, one can see that when a material approaches a (2nd order) phase transition, there will be no natural length scale in the sample (length scale --> infinity). This is the whole idea behind the renormalization group -- you keep "integrating out" the systems information on the smaller length scales and you "flow" to the description which takes into account only physics over large(~infinite) lengths which lets you study physics close to phase transitions. So it's applicable here because real world statistical systems exhibit scale invariance near critical points.
When doing fundamental physics, we would like to explain the masses of various fundamental particles. If the masses of particles are "light" then the length scale in the system becomes long and again, we develop a scale invariance. So that could be one motivation for looking at conformal invariance in particle physics. Let me be clear that the real world does NOT have conformal symmetry. (I wonder if we can make statements using that as an approximate symmetry).
In string theory, we consider a string world-sheet (2D) moving through some number of spacetime dimensions. Now, when we try to embed this sheet in spacetime, that should be independent of any coordinates we can fix on the sheet, since those are just some artificial parameters on the sheet. So that theory will have a local scale invariance (i.e. how fast you choose to vary your parameters locally) and that's why conformal invariance is talked about so much in th context of string theory. Once you reduce it to conformal field theory on a 2D sheet, the methods used are very similar to those in 2D condensed matter.
Also, the 2D conformal group has an "infinite" number of generators. I think that has relations to integrability (i.e. solvability) of these systems since you could find many constants of motion corresponding to these generators -- maybe enough to constrain your theory to be solvable. I'm not well versed with this last point and would appreciate if somebody can comment on this.