Absence of Evidence Is Evidence of Absence; It Just Isn't Proof

Defining concepts before addressing them often helps.

> Evidence (noun): the available body of facts or information indicating whether a belief or proposition is true or valid.

Absence of evidence doesn't increase your available body of facts that can validate a proposition. That's the gist of the saying.

The classic example is sampling one bucket of water from the ocean, and declaring there is no life in the oceans, because there is no life in your bucket.