The first step in this process is to be granted access to the Shared Mailbox.  The owner of the mailbox will have to approve access; have them send this approval to  

Once you have been granted access, you will need to add the account in Outlook.  To do this, click on Outlook in the upper left corner of the screen, and select Preferences.  Then go to Accounts.  Click the plus button at the lower left of the window, and select New from the drop down menu.  

In the E-mail address field, type the full email address of the shared mailbox; in this case, Then hit Continue: 

This should redirect you to a OnePass login page, where you will enter YOUR email address (not the shared mailbox email address) and password, then click Sign On:

You will now be presented with a Microsoft MFA prompt. In this case, I have opted to be sent a notification to the Microsoft Authenticator app on my phone, but you may have it set up to require you to enter a code, either from the Authenticator app or via text message:  

Once you have completed the multi factor authentication step, the mailbox should be connected. You will probably receive a prompt to allow or deny server settings. Click "Always use my response for this server" and click Allow:

Now you're done!

If you have send-as rights to the shared mailbox, you'll notice that the From field in the new message window is now a drop down menu.  Any shared mailboxes for which you have send-as rights will show up here to send from, simply select the one you want and the message will be sent from that mailbox.