To enroll Microsoft Authenticator on a replacement phone, you simply login to your Office 365 account using an alternate authentication method (if you set one up beforehand) and then enroll Microsoft Authenticator on your replacement phone, just as you did with your original phone.
The brief video below will guide you through this process:
After activating Microsoft Authenticator on your replacement phone, set it as your default method. See the article in the "Related Articles" section for details.