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I hope this question has not been asked before.

I would like to know which Ideas led (Deligne), Kazhdan and Lusztig believe, that Kazhdan–Lusztig polynomials can be expressed via intersection cohomology groups of Schubert varieties?

Is there nowadays more insight why intersection cohomology groups (or perverse sheaves) appear in representation theory? (I should mention that I know about Koszul-Duality and localisation, so if there is another viewpoint I would appreciate to hear about it)

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First I'd comment that there are quite a few questions on MO related to this one, but apparently not quite identical. (It's hard to search the site efficiently.) In any case I won't attempt a detailed answer but will rather suggest a reference.

While Kazhdan and Lusztig themselves must have interesting views on this question, Steve Kleiman (MIT) has published a detailed history of intersection cohomology along with Kazhdan-Lusztig theory. What I think is the last revised version of this article, The Development of Intersection Homology Theory, published as Pure Appl. Math. Q. 3 (2007), no. 1, 225--282, can be found here. Obviously it's difficult to sort out the history and motivation to everyone's satisfaction, but Kleiman got enough feedback along the way to be reliable in his version.

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