Not sure what you're asking. You say you've been using 365 for Mac, but then say you just downloaded the apps. Which means you can't have been using the apps previously.
Regardless, when you purchased Office 365, you should have gotten a confirmation email noting your purchase. Generally, the product is activated for you by the reseller (or directly from Microsoft) under the email address you used for the purchase.
Login to Microsoft's portal. If you don't have an account yet, create a new one and make sure to use the same email address you used for the purchase. The first thing you should see is a confirmation page showing your purchased product, similar to this:
The product key link doesn't really tell you anything you need to know. It's just the typical, long product activation key Microsoft uses. But that's not how you activate 365.
Now that you have an account and password. Launch any 365 app. It will ask for your user name. That would be the email address. The next screen will ask for your password. That's the login password you created for your MS account, not the product key.
The activation process will check your account. Assuming all was entered correctly, your 365 suite will be activated on your Mac.