It's an orthonormal basis. See my comment here[0] for more on why the forward and inverse transforms look similar (I've written e_w to mean e^iwt, but we want to think of it as a vector). The forward direction is doing a dot product with an exponential with a specific frequency to get the transform at that frequency (i.e. the function's projection/component at that frequency), which is a sum over the entire time basis. The inverse transform sums over all components/projections of the function at each frequency to rebuild the function. This is how you do an expansion in terms of orthogonal projections in any dimension.
This also explains why the forward transform has a minus and the inverse doesn't: a complex dot product sums over complexConjugate(v_i)*w_i, while the inverse/reconstruction just sums over the basis exponentials scaled by the Fourier coefficient for that frequency.
You can also prove that the sum over all imaginary exponentials is Dirac delta[1], so you could think of the inverse transform as being a dot product with delta to get the function at a specific time, and that dot product is a sum over the frequency basis.
[0] https://news.ycombinator.com/item?id=38658312
[1] https://math.stackexchange.com/questions/1343859/why-does-in...