In essence the author registers with a collection agency, and registers their songs too, which includes any ISRC codes (unique codes identifying recordings). Collection agencies then do the international collection of royalties from performance etc. You can register specifically with royalties collection agencies in other countries but I don't think there is any real advantage in this unless the collection agency you are registered with doesn't have an affiliation in that country.
Agencies have partner / aflliate agencies in other countries and essentially the foreign territory agency collects on their behalf and then sends the money to the agency you registered with.
Income comes from several streams as a songwriter. Some is dependent on coming from organizations you have a contract with, such as publishers or record company. Have a look at the music business articles on Songstuff to help too
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