exchange (an item from a store) for (another item)

To "exchange" two things means to switch them:

We exchanged business cards, but that was the last I ever heard from her.

At a store, "exchanging" an item means returning it to the store, and getting something else from them instead.

If it doesn't fit you, let me know and I'll give you the receipt to exchange it.

This is different from "returning" an item, which is when you give the item back to the store and they give you back your money.

In the U.S., most clothing stores will allow you to exchange clothes that you buy for something else. Other stores like electronics stores will also let you exchange something if it's broken. A lot of stores will also allow you to return items, though some places only let customers exchange their items for store credit. This is money that can only be used in that store.

a receipt

A "receipt" is a piece of paper (or maybe an electronic document) which shows how much someone has paid for something. You get one of these when you buy something.

A common practice at stores in the United States is to allow customers to return items to the store that they don't want, if they bring the receipt to prove that they bought the items at that store.

(clothes) fit (someone)

Say this about clothes:

Does it fit you?

You can also just say:

Does it fit?

let me know

The phrase "Let me know ___" means "Tell me ___". It's extremely common in everyday English, because "Tell me ___" can sound too demanding or angry.

Here are some examples:

Let me know when you're free and we'll set up a meeting.

Let me know when you hear from them.

If you encounter any problems, please let me know immediately.

You can see "Let me know..." a lot in business emails.