Question about relating basis of curvilinear coordinate systems.

Wikipedia gives the relationship between a cartesian and curvlinear coordinate system as
gi=(partial)x1/(partial)zi +(partial)x2/(partial)zi

Where gi is the i'th basis in the curvlinear coordinate system, x1 and x2 are the cartesian coordinates, zi, is the i'th curvlinear coordinate. Taking i as 1 or 2 in the two dimensional case.

I am having trouble deriving a proof for why this is true. Despite being a first year mathematics student i still feel that this is something within my grasp (I am currently studying vector calculus). Could someone please provide me with some geometric intuition for this identity?

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