Wikipedia claims it was invented as an ad-hoc convention by some Twitter users, and eventually it became so popular Twitter started turning @username into links.

The ~ character for home directories was an old convention that dates from the ADM-3A (1976) terminal used by some early Unix users. The keyboard on that terminal happened to have the cursor control word "Home" on the "~" key. This shorthand was adopted by shells like sh/csh and emerged in HTTP urls as /~user/ being the shorthand for a user's personal web page on a site.

Much later in history Twitter popularized the form "@user" to refer to a personal identity. I'm not sure if they invented the usage or not. This is distinct, but probably somehow cognitively related, to the use of "user@host" for email addresses after bang paths fell out of favor.

For reasons I can't quite put my finger on @user seems a much better sigil than ~user to me, so I'm not bothered that it's become popular.