Question: I've recently read about material dispersion in an optical fibre which is caused when light that enters have several different wavelengths. What I don't understand is why would wavelength affect the speed of light in the core in anyway?

Answer: 
This is because the speed of light inside the fibre depends on the index of refraction $n$:
$$
v=\frac{c}{n}\, .
$$
In turn, $n$ depends on the wavelength. It is possible to compute this dependence based on the response of the material to the electric field of the wave, which leads to the permittivity $\epsilon$ to depend on the frequency $\omega$ of the wave, and thus on $\lambda$. It is simplest to observe this experimentally.
In multimode fibre there are a number of different modes each of which has a different $\omega(k)$ relation: Typcally we will have something like 
$$
\omega(k)=\left(\frac{c}{n\_{\rm glass}}\right)\sqrt{k^2+\Omega\_{mn}^2} 
$$
in which the frequency has a lower cutoff $\Omega\_{mn}$ below which thelight cannot propagate in the fibre. The numbers $m$, $n$ label the different modes which look like standing waves in the transverse direction and propagating waves in the direction along the fibre.

This is quite different from dispersion in bulk glass where it is the refractive index $n\_{\rm glass}$ which varies with frequency. The different modes can be loosely though of as the light bouncing (by total internal refrcation) off the cladding of the fibre at different angles. If the light bounces at a lorger angle to the directon of propagation. it spends moretime going sideways and less time going in the direction along the fibre --- think of a sailboat tacking at an angle to the direction in which it wants to go -- and so makes its way along the fibre at a slower rate. 

The wikipedia article <https://en.wikipedia.org/wiki/Optical_fiber#Multi-mode_fiber>  

has some illustrations of this.