The gift box is where the items that you can't carry at the moment go, along with most items in general.
Here you can also see how many kares you are carrying and can carry, how much men you have, and how much friend points you have.
You can organize your gift box by
すべて: all
カレ: kares
アイテム・キュピチケ・ステップキー: items, tickets, keys
メン: men
ともだちpt: friend points
The pink button (まとめて受け取る) at the top takes every item on that page at once (10 items per page).
Below the item name is the date when you got the item.
The grey text (such as リボン交換のごほうびです) explains how or where you got the item.
The pink text (such as カレの所持上限を超えています) next to an item means that that item cannot be taken to due insufficient space.
Click the pink button (受け取る) to take the item.
There are limits to how much of an item you can have at one time.
1,000,000 of each items
10,000 step up keys of a rarity
9,999,999,999 men
1,000,000,000 friend points
Items in the gift box do not expire unless they are event cupid tickets or items which do have an expiration date on them, so it is recommended that you take and use them as soon as possible.