How do I make a page appear like the “Your latest posts” from homepage display options?

I didn’t get actually what you are asking about but as per i understand, i want to make you clear about setting homepage. I think you are confuse of it.

Basically, homepage setting is the place where you can set your page(home page) to a home page and a page(blog page) to blog page. When you select the homepage setting to Your latest post, instead of showing your front-page.php , home.php or index.php layout is displayed with your latest post.

If you select static and set homepage and blog page, your front page will displayed whatever is in front-page.php and whenever you go to that page, you will see your front page. And for blog page, home.php or index.php is displayed which shows your latest post even after going to page that you set for blog.

This is all about the template hierarchy of wordpress. If you want to know more,
[visit] https://developer.wordpress.org/themes/basics/template-hierarchy/