End-user instructions
You have to register for MFA (Multi-factor Authentication) once. You will be prompted for that automatically when your company administrator turns the feature on for you. Optionally you can also open this page to begin the registration: https://aka.ms/MFASetup
Look at this video how it works (thanks to Microsoft):
Recommendation
Choose to use the Mobile app, e.g. Microsoft Authenticator
Choose option "Use a verification code" to implement highest security. Otherwise select "Get message for verification".
Install Microsoft Authenticator again when you swich phone, then use this URL to register the new phone: https://aka.ms/MFASetup
Save the "App password" only if you use old/odd type of software. You have that option when you register, last in the process.