It's probably straightforward, but I would like to see the proof of the identity:
$$g_{\mu\nu}g^{\nu\alpha}=\delta^\alpha_\mu.$$
In the book 'Spacetime and Geometry' by Carroll, this identity is the motivation for calling $g^{\mu\nu}$ the 'inverse' of the metric.
Here is another way to consider it : the metric tensor $g$ is a musical isomorphism, ie an invertible map between vectors and dual vectors :
\begin{equation} g : TM \to TM^* \end{equation}
Given by the partial application
\begin{equation} X^\flat = g(X, -) \end{equation}
So that if you have a vector $X$, the dual vector $X^\flat$ acts on vectors as
\begin{equation} X^\flat(Y) = g(X,Y) \end{equation}
In coordinates, this is the raising and lowering of indices, in the case of the flat isomorphism
\begin{equation} X^\flat = (X^\mu \partial_\mu)^\flat = X^\mu (\partial_\mu)^\flat = g_{\mu\nu} X^\mu dx^\nu \end{equation}
As an isomorphism, you have also an inverse map, the inverse metric, $g^{-1}$, which is a map acting on dual vectors the way that the metric acts on vectors, ie :
\begin{equation} g(X,Y) = g^{-1}(X^\flat, Y^\flat) \end{equation}
and it also has its associated musical isomorphism :
\begin{equation} \omega^\sharp = g^{-1}(\omega, -) \end{equation}
which in terms of components acts via raising indices :
\begin{equation} \omega^\sharp = g^{\mu\nu}\omega_\mu \partial_\nu \end{equation}
The identity of the contraction of the metric and its inverse corresponds to the fact that those maps are inverse of each other, ie :
\begin{equation} (X^\flat)^\sharp = X,\ (\omega^\sharp)^\flat = \omega \end{equation}
Which leads us to
\begin{eqnarray} (X^\flat)^\sharp &=& (g_{\mu\nu} X^\mu dx^\nu)^\sharp\\ &=& g_{\mu\nu} X^\mu (dx^\nu)^\sharp\\ &=& g^{\nu\alpha} g_{\mu\nu} X^\mu \partial_\alpha\\ \end{eqnarray}
This means that for this to give us back the original vector $X^\mu \partial_\mu$, we need this to be the identity map :
\begin{equation} g^{\nu\alpha} g_{\mu\nu} = \delta^\alpha_\mu \end{equation}
Matrix multiplication of 2 matrices $g$ $h$ to $R=g\cdot h$ is defined as
$$R_{mn} = \sum\limits_l g_{ml} h_{ln}$$
This can be found in each book on linear algebra or in any handbook on math. If the result matrix is the identity $I$, one could write:
$$I_{mn} = \sum\limits_l g_{ml} h_{ln}$$
The Kronecker symbol actually is identical with the identity matrix. It is 1 if the indices $m$ and $n$ are equal (the same), and zero if they do not agree. It does not matter here if the indices are up or down. One has to keep in mind that if one works with co-variant (index down) and contra-variant (index up) indices, if the summation over an specific index is done, the index should be in one matrix up and the other matrix down. The order does not matter for tensors (attention with spinors).
Then I write:
$$\delta_m^n = \sum\limits_l g_{ml} h^{ln}$$
This does not change the algebra of the matrix multiplication, the summation still goes over the second index of the first matrix and the first index of the second matrix.
As the Kronecker symbol symbolizes the matrix elements of the identity matrix we have now in index-free formulation:
$$ I = g\cdot h $$
obviously $h$ is the inverse matrix of $g$, so I can write
$$ I = g \cdot g^{-1}$$
so we learn that $h^{ln}$ symbolize the matrix elements of the inverse metric tensor matrix $g$. For convenience we abbreviate $h^{ln}$ with the same letter, i.e. $g$, since we can it always distinguish it from $g$'s matrix elements $g_{mn}$ since the indices are up with respect to the matrix element indices of $g$ which are down:
$$h^{ln} \equiv g^{ln}:=(g^{-1})_{ln} \quad\text{whereas}\quad (g)_{ln} = g_{ln}$$
The latter makes only sense if co- and contravariant index formalism is applied.
Therefore:
$$\delta^{n}_m = \sum\limits_l g_{ml} g^{ln}$$
Finally usually when working with indices, Einstein's summation convention is applied, i.e. the summation symbol is omitted, and the summation is assumed (implicitly) on the indices which appear twice and in case of covariant and contravariant indices the summation should take place on the index (or indices) of which one appear down and the other up. In our case the only index where this occurs is $l$, we can omit the summation symbol $\Sigma$:
$$\delta^{n}_m = g_{ml} g^{ln}$$
That's it.
The use of covariant and contravariant indices is always useful if the scalar product of 2 vectors is not just:
$$ s = \sum\limits_{ij} \delta_{ij} a_i b_j = \sum\limits_{i} a_i b_i$$
but involves a non-trivial matrix for instance $g$:
$$ t = \sum\limits_{ij} g_{ij} a^i b^j = \sum\limits_{j} a_j b^j$$
In classical mechanics the first scalar product is common (except one works in a symplectic space), so co- and contravariant indices are not necessary. However, in (already special) relativity the scalar product is non-trivial, therefore the introduction of a metric as well as co- and contravariant indices are useful.
Proof of this relation depends on what approach is your textbook used to learn tensor calculus, for example, If you already defined the action of $g_{\mu \nu}$ for lowering indexes, now you can write $$g_{\mu\nu}g^{\nu\alpha}=g_{\mu\nu}e^\nu \cdot e^\alpha = e_\mu \cdot e^\alpha = \delta^\alpha_\mu$$
This proof is in Core Principles of Special and General Relativity by Luscombe:
The statement
$g_{\mu \nu} g^{\nu a} = \delta _{\mu}^{a} $
Is a convenience in tensor notation for writing out that matrix $g_{\mu \nu}$ indexed by $\mu, \nu$ multiplied by the matrix $g^{\nu a} $ indexed by $\nu, a$ is the identity matrix. This statement either can be derived if both the $g$ are known or otherwise it is a DEFINITION of one of the $g$ tensors. Most people know it as the latter.
The einstein summation convention of summing over $\nu$ above is equivalent to viewing $g_{\mu \nu}$ as a matrix and $g^{\nu a}$ as a matrix and the end result is that
$$ g_{\mu \nu} g^{\nu a} = I_d $$
where $I_d$ is an identity-like matrix (like because the $I_d$ is not necessarily square in our abstract setting, though in practice it will always be square ) indexed by $\mu, a$
Writing out these matrix identities requires defining dimensions, and inventing additional shorthand for our "identity-like" matrices. Syntactically this is a lot of work hence its' easier to just put down the kronecker delta and call it a day.
