I'm changing to a new computer.

Nice to meet you, *****!
Did you log in to your Microsoft account?

Did you download Office on your computer?

Understood
Please try these steps:

1. Go to your Microsoft account dashboard and if you're not already signed in, select Sign in. --> account.microsoft.com
2. Sign in with the Microsoft account you associated with this version of Office​​​​​​​.
3. From the top of the page, select Services and subscriptions and on that page find the Office​​​​​​​ product you want to install and select Install.

Let me know when it is done and if you have any difficulties with further setup :)

Yes
Here's how to view your product key in the Microsoft Store:

1. Go to www.microsoftstore.com. In the upper right-hand corner, select Sign in and enter the user ID and password you used to buy Office.
2. After you sign in, select your name in the upper right-hand corner, and then select Order History.
3. Locate the Office one-time purchase or individual Office app, and then select Install Office to view your product key (this doesn't actually install Office). Note that this product key won't match the product key shown in the Microsoft account, Services & subscriptions page. This is normal.
   But you may also see an option to sign in to your Microsoft account instead of entering the product key

First, may I ask you to send me a screenshot of what the screen where it asks for Product key?

You said that it asks for a product key so just send me the screenshot of this

Oh, I'm sorry

Yes, it should work without the product key. You will just need to sign to your account

You signed in on the website but you also need to do it now in the app. But you might be automatically logged in.
Did you open the Office?

Yes! 😊
If you bought it as a one-time purchase but not as a subscription

Yes, you can install Microsoft Office 2016 on multiple computers :)