Intuition for how a Fourier transform handles signals where the frequency varies with time?
Each frequency burst is equivalent to a rectangular window on an infinite sinusoid.  A rectangular window in the time domain is the same as circular convolution with a Sinc function in the frequency domain.  Thus, you end up with a spike with side humps (the spike for the sinusoid, and the humps due to the convolution with the transform of the rectangular window) for each frequency.

Then, since the FFT operation is linear, the 3 sine bursts or, equivalently, the 3 spiky hump groups, just sum additively in both the time and the frequency time domain, respectively.

You are correct in that the phase information is important for reconstruction.  The FFT phase results includes the information about the location of each (non-overlapping in your case) rectangular window for each sine burst.  Twist (or untwist) each spiky hump group's phase in the frequency domain, and you will move the burst's location (the rectangular window) in the time domain.