The proportionality constant $$k$$ (or $$k_B$$) is named after him: the Boltzmann constant. It plays a central role in all statistical thermodynamics. The Boltzmann factor is used to approximate the fraction of particles in a large system. The Boltzmann factor is given by: $\exp(-\beta E_i)$
The probability of finding a molecule with energy $$E_i$$  is equal to the fraction of the molecules with energy $$E_i$$. The average energy is obtaining by multiplying $$E_i$$ with its probability and summing over all $$i$$: $\langle E \rangle = \sum_i E_i P_i$