Why “independent evidence”?

Compare:

“The theory is correct because the theory says so.”

That is circular.

Now:

“The theory predicts X, and independent historical records show X.”

That is much stronger.

Here independent means the evidence does not merely derive from the same assumption you are trying to establish.

For example:

Theory:

Legal freedom increases labor mobility.

Independent evidence:

Census records show increasing movement of formerly tied workers into industrial employment.

The census evidence provides an independent basis for assessing the theory.