The implicit function theorem is critical in the theory of manifolds (especially that of Riemann surfaces) in showing that a subvariety of affine or projective space is actually a submanifold. I found this out bluntly after trying to read a textbook on Riemann surfaces and realizing that I didn't entirely understand it for this reason. Furthermore, the conditions of the implicit function theorem motivate the definition of a non-singular point of a variety, and in more advanced algebraic geometry, the notion of an etale map. Another important notion in algebraic geometry motivated by the implicit function theorem is that of a local complete intersection. To read an elementary account of this latter notion, see I.3 of Algebraic Curves and Riemann Surfaces by Miranda.
A good book to see how multivariable calculus and commutative algebra interact is Smooth Manifolds and Observables by Nestruev.
