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The blue lines touch the surface where they cross.
The blue lines are orthogonal (meaning, the angle between them is 90 degrees).
The lines are tangent to the surface in orthogonal planes. The planes are parallel to the zy-plane and parallel to the zx-plane. These are the same planes that the green grid lines are drawn in.
The two lines form a plane, known as the tangent plane. It is the extension of the idea of the tangent line to three dimensions.
The slopes of the blue lines are given by the partial derivatives.
A similar animation for a more complicated surface is given below.
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