We define engagement in terms of clicks. If a user clicks something on a page (i.e. doesn’t bounce), they engage with it. It is thought that you can use engagement to infer that your content is at least nominally relevant. And the deeper a user engages with a page, the more relevant they find it. The goal of web publishing is to get deep engagement.
You define deep engagement by assigning value to your links. Perhaps a click on a persistent navigation element counts as a 1 on a scale of 5, and clicking the first link in the center pane of your page counts as a 5 out of 5. Every new click adds value to the engagement.
Tuning your pages for engagement is more of an art than a science. You can have highly relevant messages but, if the design hinders the visitor from seeing them, the user might not engage. Also, referring URLs affect engagement: If a user lands on your page unexpectedly, the content might be relevant, but they are not ready or willing to engage. If you have low engagement rates, there are ways of finding out what’s wrong, but it might take a while.