The basic idea of the bandgap reference is to obtain a low temperature coefficient voltage source by cancelling the negative temperature coefficient of the diode forward voltage with the positive temperature coefficient of differential $V_{be}$. The difference of the $V_{be}$ of two different size but otherwise identical bipolar transistors is $\Delta V_{be} = k T /q \ln \eta$ (PTAT), where $\eta$ is the current density ratio, k=1.38e-23, q=1.6e-19. The diode voltage tempco is about -2mV/°C; to match that, $k/q \ln \eta $ = 2m, so $\Delta V_{be}$ = 600mV at the room temp. And the diode voltage at the room temp is also about 600mV. So the output is about 1.2V, which they say is close to the silicon bandgap voltage (1.166V at 0K). But the essential scheme does not really seem to depend on it.
Update:
Recently I watched a lecture by A. Paul Brokaw, "A Transistor Voltage Reference, and What the Band-Gap Has To Do With It", 1989. He gave the clearest explanation that I've ever seen. PTAT ($\Delta V_{be}$) and CTAT ($V_{be}$) with the right proportion add up to the constant reference, and at 0K, PTAT is 0V and $V_{be}$ is equal to the bandgap voltage.
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