In the 6th book/movie of Harry Potter Aragog the big giant spider dies. Why is this important to the story?
In addition to Major Stackings' answer, remember that Harry was able to bond with Horace Slughorn during the funeral of Aragog. Thus the funeral was a capital event : it allowed Harry to retrieve very important memories from Slughorn, which led him and Dumbledore (although the later already had suspicions) to understand Voldemort's schemes.
In the novel (and the movie), Slughorn had shown a deep unwillingness to spend any time with Harry after their brief chat about Voldemort:
‘Sir,’ Harry began, but Slughorn immediately glanced over his shoulder; when he saw that the room was empty but for himself and Harry he hurried away as fast as he could. ‘Professor – Professor, don’t you want to taste my po—?’ called Harry desperately. But Slughorn had gone.
It was only when he realised that he would have the opportunity to get his hands on some extremely valuable acromantula venom that Slughorn was willing to join Harry for a more extensive talk:
‘Touching, touching,’ said Slughorn absent-mindedly, his large droopy eyes fixed upon the distant lights of Hagrid’s cabin. ‘But Acromantula venom is very valuable … if the beast has only just died it might not yet have dried out … of course, I wouldn’t want to do anything insensitive if Hagrid is upset … but if there were any way to procure some … I mean, it’s almost impossible to get venom from an Acromantula while it’s alive …’ Slughorn seemed to be talking more to himself than Harry now. ‘… seems an awful waste not to collect it … might get a hundred Galleons a pint … to be frank, my salary is not large …’
The somberness of the funeral (reminding him of Lily Potter) and the extensive drinking session afterwards lowered Slughorn's inhibitions sufficiently for Harry to collect the true memory, something that was essential to advance the plotline.
Aragog was the only liaison between the wild spiders and Hagrid. Once he died, there was no one to forbid the forest spiders from eating Hagrid or anyone else for that matter. After Aragog passed, not even Hagrid was safe to travel near the spiders' domain. See personalities and traits on this wiki page.
As perhaps a more theoretical answer to go along with Kalissar and Richard, the death of Aragog set the stage for loss of a "beloved" character early in the novel. Though Harry had less than fond feelings for Aragog, Hagrid did, and Harry was able to feel sympathy for Hagrid. This, I would argue, was able to in part prepare Harry and the reader for the loss of another major character later in the book, and other characters in the last book. Whether intentional or not, Rowling eased the reader and Harry into loss at different points, building up our "strength" as it were to be able to better cope with loss.