I'm told that an integral of the form $$\int_{-\infty}^\infty f^*(x)g(x)dx$$ is just a generalization of the standard inner product of two vectors in $\Bbb C^n$: $$\langle v,w\rangle = \sum_{i=0}^n v^*_iw_i$$ and I do see how they are similar. Both sum over the components of the "vectors" (one just has uncountably many components).
But the integral "product" has something that the sum "product" doesn't -- a $dx$. So not only are you multiplying the components (values) of the two functions together, but you're scaling them by that infinitesimal $dx$. I see why we probably need that, if we tried to just sum $f$ and $g$ like $\int_{-\infty}^{\infty} f^*(x)g(x)$ without the infinitesimal, the integral would almost certainly blow up.
So can anyone explain why we consider the integral inner product to be a generalization of the sum inner product?
Forget $dx$. The case is the following: $C^0([0,1])$ (for simplicity) is a vector space, and
$$C^0([0,1]) \times C^0([0,1]) \rightarrow \mathbb{R}$$ $$ (f,g) \mapsto \int_0^1 fg$$
is a bilinear map which is non-degenerate, and this is the definition of inner product: a generalization of the standard inner product on $\mathbb{R}^n$ ($\sum x_iy_i$) in the sense that it also satisfies the properties: bilinearity and non-degeneracy. If you want to consider $\mathbb{C}$, you also put in the hypothesis of anti-symmetry and adjust the linearity hypothesis to only hold on the first variable.