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My answer is maybe more to praise the glory of perverse sheaves than the decomposition theorem exactly, but bare bear with me. To appreciate the theorem, I'd say first get a sense of what Hodge theory for smooth projective algebraic varieties says: hard Lefschetz etc. (already the fact that the proofs you're likely to see involve harmonic forms and analysis should convince you this is serious stuff). Then try to get a sense of what it means to understand this theorem in families, where things like Hodge filtrations start to appear.

Finally despair of what it might mean to even consider this picture if the "family" you were looking at was just a projective morphism $f\colon Y \to X$, where $Y$ is smooth: the local systems you need for the families version of Hodge theory break down. However, enter perverse sheaves, as sort of singular local systems, and the decomposition theorem says the whole picture is miraculously saved. Viewed this way I think you get a proper sense of how amazing the theorem (and the discovery of perverse sheaves) really is.

P.S. This answer is a poor attempt to convey what others have told me: a better attempt is made in de Cataldo and Migliorini's articlehttp://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf

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My answer is maybe more to praise the glory of perverse sheaves than the decomposition theorem exactly, but bare with me. To appreciate the theorem, I'd say first get a sense of what Hodge theory for smooth projective algebraic varieties says: hard Lefschetz etc. (already the fact that the proofs you're likely to see involve harmonic forms and analysis should convince you this is serious stuff). Then try to get a sense of what it means to understand this theorem in families, where things like Hodge filtrations start to appear.

Finally despair of what it might mean to even consider this picture if the "family" you were looking at was just a projective morphism $f\colon Y \to X$, where $Y$ is smooth: the local systems you need for the families version of Hodge theory break down. However, enter perverse sheaves, as sort of singular local systems, and the decomposition theorem says the whole picture is miraculously saved. Viewed this way I think you get a proper sense of how amazing the theorem (and the discovery of perverse sheaves) really is.

P.S. This answer is a poor attempt to convey what others have told me: a better attempt is made in de Cataldo and Migliorini's article http://www.ams.org/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf