I think it is just the language works. In Spanish it is the same: "¿Cómo te llamas?", but we could use "¿Cuál es tu nombre?" (almost literally, "What is your name?"). The latter would be more familiar to English speakers, but the former makes sense on its own, both in Spanish and Italian, because you're kind of asking someone "what name do they go by".
Come= what, ti=yourself, chiami=you call
"What, yourself, you call" or in better english "What you call yourself". And in everyday english "What is your name".
("Ti" can also mean "you" or "to you", - but as in this sentence it is "you" who "call", and the action of calling goes back on "you" (a reflexiv construction) it should be translated as "yourself" here.)
That's because of differences in the languages themselves (see my other answer elsewhere on this same page). A more direct transation of "Come ti chiami" would be "how are you called" or "how do you call yourself".
But in English you don't say that. You ask: "what's your name?".