[Edit: This answer seems to fit the title of the question, though not the actual question in the body.]
Resoluion of singularties in algebraic geometry seems like a good example. Hironaka's original proof was over 200 pages and hard to understand:
"Even A. Grothendieck [in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1, 7--9, Gauthier-Villars, Paris, 1971; MR0414283 (54 #2386)] admitted openly that he did not completely understand Hironaka's proof."
That quote is from Dan Abramovich's Math Review of the book Lectures on resolution of singularities by Kollár; the review goes on to say
"One can [nowadays] devote a few weeks in a first course on algebraic geometry to give just a complete proof of resolution of singularities in characteristic 0 (Chapter 3 of the present book, which is largely self-contained)."
I know almost nothing about this topic, but some names I know associated to the various approaches to simplification of Hironaka's proof are Bierstone, Milman, Encinas, Villamayor, Hauser, Cutkosky, Włodarczyk, Kollár. Please tell me any I missed!