The wikipedia diagram at https://en.wikipedia.org/wiki/Gaussian_correlation_inequalit... helps gain an intuition for the Gaussian correlation inequality. The diagram shows a circle and a rectangle centered on the origin.

The number of random dart throws in the intersection of the shapes is greater than or equal to the number in either individual shape.

Well, sort of. The wikipedia page says "the proportion of the darts landing in the intersection of both shapes is no less than the product of the proportions of the darts landing in each shape."