The article, and quite a few posts here, describe the way they understand the Fourier Transform as the way to understand the Fourier Transform. For it to be intuitive depends on who is trying to understand it. Getting that out of the way, this is how I find the Fourier Transform intuitive (using pseudo-code instead of math notation to make it a bit verbose and emphasize the steps):
fourier_trasform(signal sig(t), frequency freq):
let sinu(t) = sinusoid with frequency freq
let mult(t) = sig(t) * sinu(t)
value = integral of mult(t) from -infinity to infinity
If the input signal sig(t) has the same frequency as the sinusoid sinu(t), then integrating mult(t) over infinity will give an infinitely large value, and that case is handled better by the Fourier Series.
If the input signal sig(t) has no relation to the frequency of the sinusoid, then integrating mult(t) over infinity will give zero.
If the signal has a component with the required frequency, it will kind of resonate with the sinusoid and give a non-zero value. The value then depends on the magnitude of the signal and to how much it "resonates" with the sinusoid.
When you do this for a range of different frequencies freq, but using the same signal sig(t), you can plot how much the signal sig(t) resonates with all frequencies, and that plot is the plot of the Fourier Transform.
By the by, one of the things that always tripped me up about the DFT was understanding that for signals that are already sums of sinusoids, the discrete sum gives exactly the same answer as the integral would. Remember the trapezoidal rule for integration? It turns out it's exact for trigonometric polynomials sampled at or above the Nyquist frequency.
Now, to find a life sciences journal to publish that in...
If the input signal sig(t) has no relation to the frequency of the sinusoid, then integrating mult(t) over infinity will give zero.
If the signal has a component with the required frequency, it will kind of resonate with the sinusoid and give a non-zero value. The value then depends on the magnitude of the signal and to how much it "resonates" with the sinusoid.
When you do this for a range of different frequencies freq, but using the same signal sig(t), you can plot how much the signal sig(t) resonates with all frequencies, and that plot is the plot of the Fourier Transform.