E.164 is the international numbering format for phone numbers: a plus sign, the country code and the national number, up to 15 digits, with no spaces, zeros or punctuation.
E.164 is the ITU-T recommendation that defines how a telephone number is written internationally: +, country code (1 to 3 digits), then the subscriber number, for a maximum of 15 digits. +14155550142 and +447700900123 are E.164; 07700 900123 is not.
APIs use E.164 because it is unambiguous: the country, and therefore the route, the price and the sender rule, can be read from the prefix. Local formats need a default country to be interpreted, and the same digits can mean different numbers in different countries.
At SMSMeteor: the API accepts E.164 only and rejects anything else with invalid_number before billing; the panel importer converts local formats when you tell it the default country.