Is it possible to display a Mailchimp campaign (the newsletter, not a link) from my Drupal 7 website?
Outside of linking or iframe embedding, there isn't a way of simply displaying the content as, say, a Drupal page. There is actually a reason for this.

First, the content of the email is already an entire webpage: it (should) have an &lt;html&gt; tag and it will likely have &lt;head&gt;, &lt;style&gt;, and &lt;body&gt; sections. So it actually makes plenty of sense to just include it as an iframe directly on a page since it will need it's own space to format the content properly.

But let's say you decide to use the /campaigns/content.format API call to grab the actual content of the campaign. You'd likely still need to strip away the CSS and other HTML from within the content to make it display within your page nicely.

And with regards to synchronization, it only syncs from Drupal to Mailchimp. You'd need some other way of grabbing those campaigns if you wanted to see them in Drupal. This is especially useful because Mailchimp is used by many who have a single account + multiple clients.