One nice family of examples is brought to us from thermodynamics.
For example, the ideal gas law states:
$$ PV = nRT $$
where $P,V,n,T$ are generally variables which describe the pressure, volume, number of particles and temperature of a gas. We calculate the total differential:
$$ VdP+PdV= RTdn+RndT $$
This equation of differentials contains a multitude of possible interpretations. One beautiful truth, at the outset, this total differential contains all those choices. In the absence of further constraints, any one of the four variables here can be taken as the dependent variable which is given by a function of the three remaining variables. The possibility of taking a given variable as dependent simply amounts to the algebraic freedom to solve for the differential of that variable. For example,
$$ dP = \frac{1}{V}(-PdV+RTdn+RndT) \ \ (\star)$$
Clearly $V=0$ is troubling, we can only expect to find $P = P(V,n,T)$ in some set where $V \neq 0$. Of course, this is not really an impediment here as negative volumes are hard to find. In any event, we can read the partial derivatives of $P$ with respect to $V,n,T$ directly from $\star$,
$$ \left(\frac{\partial P}{\partial V}\right)_{n,T} = \frac{-P}{V}, \ \ \
\left(\frac{\partial P}{\partial n}\right)_{V,T} = \frac{RT}{V}, \ \ \
\left(\frac{\partial P}{\partial T}\right)_{n,V} = \frac{Rn}{V} $$
The notation above is perhaps redundant given the context, however, in general in applications where there are multiple interpretations possible for a set of variables it is nice to be able to explicitly indicate which variables are taken to be independent.
Generalizing a bit:
This example is not hard enough to really appreciate the method of differentials as it explains partial derivatives for sets of variables. Essentially the same analysis is possible for several equations which govern some set of variables. In general, the meaning of something like $\frac{\partial w}{\partial x}$ is ambiguous. The value depends on the set of variables which is taken as independent as well as the defining equations (constraints) for the problem. What justifies this formal method? It's the implicit function theorem. Thus, if you play with these differentials a bit and gain a better sense of how the Jacobian can be used to linearize a mapping then the implicit function theorem is not at all abstract. It's simply the condition needed to solve a given linearization for certain variables in terms of the remaining variables.
We can be more greedy, the contraction mapping principle paired with a Newton's method type argument even gives a sequence of functions which converges to the implicit solution in a particular manner. So, the intuition I sketch above is just the first step in understanding this deeper result. Edwards' Advanced Calculus is once source for the implicit and inverse mapping theorems paired with the contraction mapping based solution sequence results.
There is a better answer to give here. So many of the interesting theorems ultimately rest on the implicit function theorem. In some sense, those theorems are the real triumph of the implicit function theorem.