Well, your blog page is not one particular post/page, it is rather a template which populates a list of posts, and as such, it cannot have any custom fields associated to it.
Where exactly did you input the custom field value that you expected would show up on that page?
Edit: Your description is in the header, so you can simply:
Create a standard page called 'Blog'. Then go to
wp-admin -> Settings -> Reading, and change the 'Posts page' value to your 'Blog' page. You may need to adjust your menu to reflect this change.
Create a custom template for your Blog page. It should pretty much be a copy-paste from
content.php, if your theme is similar to the Twentytwelve theme. If it isn't then you'll need to investigate which template is responsible for populating your list of posts. Codex has many useful guides, most useful for you in this case is probably Pages.
Edit your Blog page from #1 to make it use your new template from #2 (
Edit Page -> Page Attributes -> Template drop-down).