To verify phone numbers in Python with the Twilio Lookup API, developers need to install the Twilio Python library and set their Account SID and Auth Token as environment variables. They can then use the `client.lookups.phone_numbers()` method to check if a number exists and fetch additional information such as carrier details or formatting. If an invalid phone number is provided, the API returns a 404 response, which can be caught by the Python library and used to validate phone numbers. A custom function can be created to handle this validation, returning `True` for valid numbers and `False` otherwise.