The material index of refraction does fall at longer wavelengths.
From Fourier analysis, we know that every pulse with limited time duration contains a non-zero spread of frequencies in its spectrum. Practically, most optical sources will produce a wider spread of frequencies than the Fourier limit allows.
Therefore, if the propagation speed depends on the frequency, some of the energy in a pulse will arrive at the receiver "early" and some will arrive "late". The pulse will be spread out. This is what we call dispersion.
Material dispersion is not the only source of dispersion. There is also an effect from the waveguide geometry itself. "Chromatic dispersion" is the combination of the material and waveguide dispersion effects. At some particular frequency, these two effects can cancel each other out giving a wavelength with approximately 0 chromatic dispersion:
why is dispersion proportional to wavelength?
Dispersion tends to increase with wavelength, but not in a proportional way, as you can see from the figure above.