What does "breed sth. out of sth." mean?

Get rid of it.

When you breed an animal, you choose a mother and father with certain traits and get them to have children. Then you choose a mother and father from the children that have this trait.

By doing this, you can get rid of some traits. For example if you don't dogs that have spots, you will "breed" dogs with no spots. You will pick a mother and father dog that don't have spots to make puppies. Of these puppies, you will select only the one without spots to have more puppies. This way, dogs with spots won't get the chance to continue whatever genes make spots, and the "spotted" gene will likely not exist after many generations of this.

The idea here is that our "conscience", as a human trait, is connected to a gene that is passed from a mother and father human to their children. If you only let humans without a strong conscience to have children, the humans with the "conscience" gene will be the only ones passing down their genes.

The theory is that if only humans without a strong conscience are having kids, eventually there will be a generation of humans without whatever gene responsible for "conscience".

For the record, I think this idea is hogwash for a couple of reasons.

your answer provides a very inspiring insight. Too many laws have made people lose their "conscience" so that their children will not inherit the conscience gene from them. is it what you mean?

She is annoyed by too many laws that are seem to make any sense so she jokingly, I think, of breeding "conscience" gene out.

In your opinion, are those laws the equals of the "conscience" or the opposite of "conscience"?

It is a joke of the writer. Conscience means a person feels that this is the proper way so feeling that some people may not have conscience they go on and make laws to follow his way of behavior. It is annoying to the author that so many laws are made and suggests weeding out the 'conscience' gene.