Multidimensional Fourier transform: inner product in exponential?

I have a basic and not very deep understanding of continuous and discrete Fourier transforms in one dimension.

In a multidimensional Fourier transform, the exponent of $$e$$ includes the inner product of the arguments of the original function and the transformed function. I’m having trouble finding an explanation of why this is the right way to extend the idea of a Fourier transform to more than one dimension. What is the intuition, or perhaps a more formal reason for this idea? In one dimension, the original function’s and transformed function’s arguments are multiplied, and using an inner product in a multidimensional case is one natural extension of that idea, but that’s not enough to justify it.

I see that using the inner product in the exponential means that the integral or sum (for a discrete Fourier transform) is over a product of $$\cos x_i + i \sin x_i$$ sums, but I am not sure why that makes sense, or even whether that’s a useful way to think about it. (I can picture waves in two real dimensions, and multiplying one-dimensional wave equations kind of feels like a good way to represent that, but I still don’t have a clear understanding.)

Pointers to texts as well as explanations here would be welcome.