This questions gets to the heart of the tragedy of the book. Mockingbirds, as defined by the book, are people who only do good. Therefore, to harm one is a sin. To use the words of Miss Maudie:

“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.”

So, how does society kill them? First, we see that Maycomb kills Tom Robinson, an innocent and good man, by its blind racism. Atticus points out that there is a disease in the town that turns reasonable people mad when race is involved. Here is what Atticus says:

“Right. But do you think I could face my children otherwise? You know what’s going to happen as well as I do, Jack, and I hope and pray I can get Jem and Scout through it without bitterness, and most of all, without catching Maycomb’s usual disease. Why reasonable people go stark raving mad when anything involving a Negro comes up, is something I don’t pretend to understand..."

Second, false religious beliefs can also harm mockingbirds, such as when Boo parents were excessively religious. They harmed their son, and so he became a recluse and dysfunctional person. Here is what Miss Maudie says about the Radleys:

Miss Maudie said: “Foot-washers believe anything that’s pleasure is a sin. Did you know some of ‘em came out of the woods one Saturday and passed by this place and told me me and my flowers were going to hell?”

Boo grew up in that type of environment. This caused him great pain. Fortunately for him, good people could see this. In the end of the book, the Finches and Heck Tate protect him. 

In conclusion, society kills its mockingbirds in various ways. Sometimes it is though blind racism, other times it it is through excessive religious beliefs, and other times it is through inaction, that is, not standing up for what is right. 

How does society destroy its "mockingbirds"?

Society destroys its mockingbirds in a few ways in the book.

First, society destroys it mockingbirds by having blindspots. The power of a blindspot is that you do not know you have it. When a society has a blindspot, it can be very detrimental. When we take this insight and apply it to Maycomb, we see that racism is a blindspot. The very air Maycomb breathes is filled with racism. Even the missionary society does not know that their views of race belie what the Bible says. Here is another example. The Cunninghams from Sarum would have killed Atticus, even though Atticus helped them. On account of things like this, Tom Robinson was destroyed. 

Second, not too many people have true courage. One underlying point of the novel is the true nature of courage. True courage is to stand up for what it right even if you will lose. Atticus demonstrates it, but the problem is that there are not enough people like Atticus. The outcome once again is that mockingbirds perish.