How optical sources having narrower spectral width gives higher data rates?
You're probably missing one thing : the transmission medium.

Optical sources for high bandwidth communications normally use glass as the transmission medium, in the form of optical fibre. Now the glass  is carefully chosen, and I expect one of the factors they optimise for is low dispersion, yet I don't believe the dispersion is 0. 

(EDIT : confirmed by a commenter : dispersion is deliberately engineered, to mitigate non-linearity by dispersing the mixing products).

Now in a dispersive medium, different frequencies travel at different velocities. Thus a pulse will smear out as it travels down the fibre, according to the different propagation speeds of its spectral components. The receiver will see a sharp pulse after a 1 km fibre, a broader pulse at 10 km, and broader still at 100 km and so on.

Thus the pulse width depends on the dispersion of the medium, the link length, and the frequency spread of the signal. And that pulse width determines the usable data rate.

A narrower line width reduces pulse broadening due to dispersion, enabling higher data rates for a given link length using a given dispersive medium.

(Improvements to this answer from an electro-optics specialist are welcome).