In The Iliad, Homer often describes the warriors rushing to strip the dead of their armor, but these same men are described as wearing armor. I assume that this is based on reality, and I'm wondering if the armor was to bring back as a trophy, or simply to have a spare set for themselves?
The purpose was prestige. Captured armour brought prestige to its new owner in a number of ways:
It was proof of a kill in combat. Since social status in Early Greece seems to have been determined to a large extent by one's ability to fight, it was very important to be able to show proof of this. Nothing boasted of your prowess like a pile of captured armour. Indeed, in one surviving fragment of his work, the Archaic poet Alkaios of Mytilene described the walls of his hall, shining in bronze and white (presumably linen) from all the stuff he'd stripped off his dead enemies. We should expect that all homes of wealthy Greeks would have such spoils prominently on display.
Armour was precious. Coinage was not invented until the 6th century BC; before that, all trade was barter in one form or another, and much took the form of the exchange of luxury objects. Big chunks of worked bronze, in the form of helmets, chestplates and greaves, were tremendously prestigious and valuable objects, and it would be a tragic waste to leave them on the battlefield for the enemy to claim back.
Elite networks were maintained by a system of gift exchange. Guests to a lord's household would be offered precious things as a mark of respect; in return, that lord would expect precious gifts if he ever visited you. Networks of patronage, too, were established by powerful lords favouring particular warriors with precious gifts. Obviously, any man of means required a steady supply of valuable items (usually metal objects, but also cloth, ivory and the like) in order to sustain his social status. Capturing enemy armour was an important way to supplement one's income and extend one's ability to operate in high circles.
These reasons explain, not just the practice of stripping armour from fallen men, but also the common scenes in the Iliad and in vase painting of men fighting over fallen warriors. Indeed, it is these aspects of social status in Early Greece that drive the plot of the Iliad in general; Achilles feels slighted by Agamemnon's decisions regarding the distribution of the spoils of war. In the Classical period, Athenian audiences apparently still understood the shame felt by Aias (Ajax) when he lost the competition for the armour of Achilles to Odysseus, which drives the plot of Sophokles' tragedy Aias.