Without anonymity, one risks real-world harassment, stalking, and political, social, or economic consequences for saying things one's society, government, or maybe even employer doesn't like.
Journalists have been murdered for reporting the actions of gangs and corrupt governments. Had they been anonymous, perhaps they'd still be alive.
LGBT people risk being attacked when their identities are revealed online. Many other groups throughout history have been targeted because of their identity -- a prime example being Jews during the Holocaust, when census data was used to find them an round them up.
People will also be hesitant about speaking about wrongs committed by police, corrupt governments, or their employer when they believe their identity is known, for fear of reprisal.
As the ACLU points out: "The right to remain anonymous is a fundamental component of our right to free speech, and it applies every bit as much in the digital world as it does in the physical one."
https://www.aclu.org/issues/free-speech/internet-speech/onli...