Why do you need a carrier or service provider ?

A carrier is a Mobile operator or service provider, who provides the communication service between the phones.
If a person A calls to a person B, the call reaches the carrier provider , say for example AT&T.
Now AT&T gets a request from the person A Mobile number, checks the destination number and carrier provider. If same carrier provider it transfers the call to the person B phone number and hence a connectivity is established between A and B. If the carrier of B is different, it transfers the call to that different carrier provider and in turn the provider transfers calls to person B. Now  A and B will be connected.