Why does Atticus say you should not shoot a mockingbird?

The mockingbird represents a sense of innocence. It merely sings and brings joy to people.

You can check out the quote here,

"Atticus said to Jem one day, 'I'd rather you shoot at tin cans in the back yard, but I know you'll go after birds. Shoot all the blue jays you want, if you can hit 'em, but remember it's a sin to kill a mockingbird.'

That was the only time I ever heard Atticus say it was a sin to do something, and I asked Miss Maudie about it.

'Your father's right,' she said. 'Mockingbirds don't do one thing but make music for us to enjoy. They don't eat up people's gardens, don't nest in corncribs, they don't do one thing but sing their hearts out for us. That's why it's a sin to kill a mockingbird.'"

The mockingbird also comes to represent innocents like Boo Radley and Tom Robinson.