This functionality lets readers identify the author of a post by displaying the name, Google+ profile picture and a small description of the author below the post. Clicking on the Author's name, takes the reader to the Google+ profile of the author. This without doubt, will help the author in getting more Google+ followers.
Below is a screenshot of my profile below a blog post on of my blogs:
To display your profile under your Blogger posts, you must have replaced your Blogger profile with Google+ profile.
Thereafter, you can log in to your Blogger dashboard, go to "Layout" and click the "edit" link of the "Blog Posts" gadget. You should see a new window similar to the one below:
==> Tick the "Show Author Profile Below Post" box as shown in the screenshot. Also ensure you ticked the "Post by..." box.
==> SAVE your settings and view your blog posts.
If you have a customized template and you don't see the author profile below your blog posts, you may need to reset your Blog1 widget in the template editor for it to be added.
How To Reset Blog1 Widget Template
==> Go to "Template" > Edit HTML
==> Click the "Revert widget templates to default" tab
==> Select Blog1 from the list of widgets and click the "revert selected widgets" button.
==> SAVE your template and view your blog posts.
I hope it works for you.
NOTE: To edit the bio displayed in your Blogger author profile, go to the "About"section of your Google+ profile and edit the "Introduction".