The custom of changing one's name to one's husband seems to only exist in those countries whose legal systems developed from English Common Law: the UK, the US, Australia, New Zealand, Canada, etc. Italian and Latin American women, for example, don't change their names.

It comes from the old idea that, in English Common Law, when a woman got married, she ceased to exist legally as a separate person and became the same person as her husband, legally. Which is why a woman used to not be able to testify against her husband in court, because legally, she was testifying against herself (since she is her husband), and you can't be made to testify against yourself in court.