What is the intuitive explanation of the Inverse Function Theorem (and generalized to multiple dimensions)?
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$\begingroup$The moral story of all differential calculus is this:
Differentiable functions behave locally like their linearizations.
The degree to which this statement holds can be quantitatively formulated, for example in terms of error estimates for first-order Taylor expansions. It also manifests qualitatively; for an easy example, if $f:\mathbb{R}\to\mathbb{R}$ has a linearization $\ell_p(t) = f(p) + f'(p)(t-p)$ which is increasing (i.e. $f'(p)>0$), then $f$ is also locally increasing.
The moral of the inverse function theorem goes like this. First, let $f$ be differentiable on a neighborhood of $p$, and suppose for simplicity that $f(p) = 0$, so that the linearization of $f$ is $f(x) = D_pf(x-p)$. Then:
If a function $f$ is differentiable on a neighborhood of a point $p$ and its linearization $D_pf$ at $p$ is invertible, then $f$ is locally invertible. Moreover, the inverse of the linearization $(D_pf)^{-1}$ is precisely the linearization of the inverse $f^{-1}$ of $f$ (which we know to be locally defined).
So all we're saying is that invertibility is one of the properties that a function $f$ can copy from its linearization $D_pf$. But because a given linear approximation is only accurate (i.e. has small error) in a small neighborhood of the point of approximation, and you can only expect $f$ to behave like $D_pf$ when the approximation is accurate, you can only conclude that $f$ is invertible on a small neighborhood.
$\endgroup$ $\begingroup$If you have $n\geq1$ independent variables $x_k$ and $n$ functions $$f_i:\quad U\to{\mathbb R},\quad (x_1,\ldots, x_n)\mapsto y_i:=f_i(x_1,\ldots, x_n)\qquad(1\leq i\leq n)$$ defined in a neighborhood $U$of ${\bf0}$, whereby ${\bf f}({\bf 0})={\bf 0}$, then under certain "technical assumptions", this setup defines the $x_k$ as functions of the $y_i$: There are $n$ functions $$g_k:\quad V\to{\mathbb R},\quad (y_1,\ldots, y_n)\mapsto x_k:=g_k(y_1,\ldots, y_n)\qquad(1\leq k\leq n)$$ defined in a neighborhood $V$of ${\bf0}$, whereby ${\bf g}({\bf 0})={\bf 0}$, such that $$y_i:=f_i(x_1,\ldots, x_n)\quad(1\leq i\leq n)\quad\Leftrightarrow \quad x_k:=g_k(y_1,\ldots, y_n)\quad(1\leq i\leq n)\ .$$
The essential point of the theorem is that it guarantees the existence of such functions $g_k$ and their properties even if you are not able to solve the system $$y_i=f_i(x_1,\ldots, x_n)\qquad(1\leq i\leq n)$$ explicitly for the unknowns $x_k$.
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