Add Read More in My Posts is the very popular button you saw in someone's summary post. In some WordPress themes, the blog templates are designed for posts summary. If your theme wasn't with that designed, don't worry, you can insert a Read more break in your post where you want to showed in the posts' summary.
Add Read More in My Posts - Open one of your old posts or Add New Post
And in your post editing area you will see this line.
Go view your post then you will see the result as mine below.