Here are some examples:
* One of a pair of shoes: You have two shoes, but you are only talking about one of them.
* One of a pair of earrings: You have two earrings, but you are referring to just one.
* One of a pair of socks: You have two socks, but you are talking about only one.
Essentially, it emphasizes that the item is part of a larger set, but you are specifically focusing on just one.