Intuitive explanation for metric inverse definition

Let $M$ be an riemannian manifold, $p \in M$ a point and $T_pM$ the tangent space at $p$. The metric tensor $g$ at a point $p$ is an inner product $g_p : T_pM \times T_pM \to \mathbb{R}$. Since it is an inner product in $T_pM$, we can define a linear transformation $\hat{g}_p : T_pM \to T_pM^*$, where $T_pM^*$ is the dual vector space of $T_pM$, by $\hat{g}_p(v) = g_p(v,.)$. The non-degeneracy of $g_p$ implies that $\hat{g}_p$ is an isomorphism between vector spaces.

In coordinates, you can check that $g_{ab}$ at the point $p$ is just the matrix that represents the map $\hat{g}_p$. Therefore $g^{ab}$ is the matrix representation of the inverse map $\hat{g}^{-1}_p$. When we use the metric tensor to raise or lower indices, we are basically applying the maps $\hat{g}$ and $\hat{g}^{-1}$ to vector and covector fields.

For last, when you write $g_{ab}$, the label of the index don't really matter as long as it is consistent inside a summation. Since the symbol $\delta^{a}_b$ is just the identity map in coordinates, $g^{db} \delta^{a}_d$ is just changing the label of the index from $d$ to $a$. This is really useful for tracking how the indices are changing after a big sum.

Also, you copied the example wrong, it's $g^{ab} = g^{ac}g^{bd}g_{cd}$, otherwise, we would have $g_{ab} = g^{ab}$ which is rarely the case.