“This might sound sexist, but I do believe a woman should stay home and take care of the children.”

You're in a discussion with some friends about women who are focused on their careers. You hold pretty traditional ideas about women working. You want to state your opinion, but you know that some of the people listening won't agree with it. You say:

This might sound sexist, but I do believe a woman should stay home and take care of the children.

this might sound (adjective), but

Use this phrase to prepare listeners for something you're going to say. Common adjectives to use in this expression are:

This might sound crazy, but what if we opened the show up to the general public and sold tickets to it?

This might sound harsh, but I say we kick him out of the apartment if he's late with another payment.

This might sound stupid, but can't we just go up to him and ask him directly?

So you use this kind of phrase by thinking of what negative adjectives people might think of, and then saying "This might sound ___, but..." When you do this, the listener can't complain as much about what you said because you've already admitted that it might be sexist, crazy, harsh, and so on.

(a) sexist (idea or statement)

Being "sexist" means that you think that men and women can't or shouldn't be allowed to do the same things. In the U.S., the "official" position of the government and the media is that men and women should be equal. A lot of individuals have different opinions, but these people can be called "sexist".

People usually call each other "sexist" for thinking that women shouldn't be allowed to do the things that men do. It's uncommon for someone to be accused of being "sexist" against men. You may hear a man call a woman "sexist" as a joke sometimes, though.

I believe (clause)

You can say "I believe ___" to express an opinion that you hold deeply. Use this to talk about political, religious, philosophical, or moral ideas:

I believe that God is looking out for all of us.

I believe that every American should have the right to own a gun for the protection of his or her family and country.

stay home and take care of the children

This whole phrase is often used together to describe what a housewife (or, recently, a house-husband) does.