You simply have to download the Monimoto app on your new phone and log in using the same log in method which you used when you first created your account:
Sign-in with Phone (this requires you to have access to the phone number you used when you first created your account due to SMS short code verification)
Sign-in with Google
Sign-in with Apple
Make sure you use the same method as your original Sign-in otherwise it will create a separate account. Each Sign-in method creates separate accounts from each other.