In Munchkin, what is the logic behind the rule of carry and equip items? What is the difference between carry and just keep the card in your hand? It is not simplier keep in hand and not use the carry rule?
Further to what the other answers have pointed out, you can only trade items that are in play, not items in your hand. Therefore you must first (legally) play an item to the table, either equipped or carried, before you can trade it with another player.
As one can only play an item (Update - other than one-shot items, which can be played from your hand when in combat.) from your hand either upon receiving it or during your turn when not in combat one might also play an item as carried in order to trade it during another player's turn, before your turn comes around again.
You equip items so you can get their benefits.
But there are restrictions to what you can equip. You can't equip items for which you don't have the required free body slots (e.g. hand). You can't equip items for which you don't meet race and class requirements. You can't equip "use only once" items.
You keep items you can't equip in your hand to protect them from traps and thieves, and to hide what they are from other players.
But there are limits to one's hand size.
You carry items you can't equip and won't fit in your hand to avoid having to discard them, or worse, having to give them to another player through the charity rule.
Note that you might hold back from equipping an item in your hand if you think you might lose it (e.g. to a thief, especially one at level 1), but there's virtually no reason not to equip something you can equip if you're currently carrying it.
If it prevents you from having more than 5 cards in hand, you avoid the charity rule, requiring that you discard down to 5 cards in hand by donating the remainder to the lowest level player or the discard pile, at the end of your turn.
The ability to carry items, allows you to build a stock of items beyond those you are able to equip, whether restricted by class limitations or available body slots. For example, short an item providing additional hands, you only have two; you can only equip 1 two-handed item, or 2 one-handed items.