The Fourier Transform, explained in one sentence (2014)

3Blue1Brown's video is the best explanation I've ever seen.

That's a question of why Fourier transforms are important though, not just how they're defined and computed. The next-level answer is presumably that sinusoids (or complex exponentials in general) are the eigenfunctions of general linear time-invariant systems, i.e. that if the input to an LTI system is exp(j*w*t), then its output will be A*exp(j*w*t) for some complex constant A. Some other comments here already alluded to that, noting that sinusoids are good for solving linear differential equations (which are LTI systems), or that the sum of two sinusoids of the same frequency shifted in time (which is an LTI operation, since addition and time shift are both LTI) is another sinusoid of that same frequency.

LTI systems closely model many practical systems, including the tuning forks and flutes that give our intuition of what a "pure tone" means. I guess there's a level after that noting that conservation laws lead to LTI systems. I guess there's further levels too, but I'm not a physicist.

That eigenfunction property means we can describe the response of any LTI system to a sinusoid (again, complex exponential in general) at a given frequency by a single complex scalar, whose magnitude represents a gain and whose phase represents a phase shift. No other set of basis functions has this property, thus the special importance of a Fourier transform.

We could write a perfectly meaningful transform using any set of basis functions, not just sinusoids (and e.g. the graphics people often do, and call them wavelets). But if we place one of those non-sinusoidal basis functions at the input of an LTI system, then the output will in general be the sum of infinitely many different basis functions, not describable by any finite number of scalars. This makes those non-sinusoidal basis functions much less useful in modeling LTI systems.

For me, the most intuitive explanation of the Fourier transform is geometric and is as follows:

If:

    . you have a vector space equipped with a dot product - say R³
    . you have a  specific orthonormal basis of said vector space - say (i, j, k) in R³ 
    . you have a vector V - say V ∈ R³
    . you want the coordinates of V in that basis
    . then all you have to do is take the dot product of V with each member of the basis (i.e. you *project* V onto each basis member).
    . in other words: V = Sum_over_basis_members[ individual_basis_member * Dot_product[V, individual_basis_member] ]

And magically, that's all the various Fourier transforms are:

   . the discrete Fourier transform (both signal and basis are discrete and finite, dot product is the usual suspect)
   . the Fourier series (basis is discrete and infinite, signal is a continuous periodic function, dot product is the continuous version of the standard dot product, i.e. an integral)
   . the Fourier transform (basis is infinite and continuous, signal is any "well-behaved" function, and the reconstruction of the "vector" involves a double integral, one for the dot product and one for the summation over basis terms)

For example, for the Fourier series:

    . The "Vector" is any well-behaved periodic function (don't recall exactly what well-behaved entails, but not the most important part of the story anyways)
    . The vector space is the set of well-behaved periodic functions
    . The basis is the family of functions Exp[2*π*i] where the parameter i = {0, 1, 2, 3, ..., infty}
    . The dot product is the cross-correlation between the function and a basis member: i.e. Integral[x=0, x=2*π, function*basis_member]
    . You rebuild the function as:

               function = Sum_over_basis_members[ basis_member Dot_product[function, basis_member] ]

And that's it.

Kind similar is this demonstration on a record player:

https://youtu.be/mRi23ueU7Zk