How to interpret that "covariance measures linear dependence"?

Question

I know the formula of Cov and relevant calculations. But I am not sure that I grasp the following sentence:

Covariance measures the amount of linear dependence between two random variables. (From Introductory Econometrics by Wooldridge)

Why can we say it measures linear dependence, rather than non-linear relationship?

Answer 1

I asked this question in the class. The professor provided one explanation. He said because the formula is just $X-\mu_X, Y-\mu_Y$, it's not $(X-\mu_X)^2, (Y-\mu_Y)^2.$

Is it the true reason?

Answer 2

Sometimes a figure is worth 1000 words:

Covariance = 0 on the left, 0.9 on the right: